890:
6570:
364:
3858:
4770:
22:
885:{\displaystyle {\begin{aligned}P(r)&=\sum _{i=1}^{n}P\left({\text{applicant }}i{\text{ is selected}}\cap {\text{applicant }}i{\text{ is the best}}\right)\\&=\sum _{i=1}^{n}P\left({\text{applicant }}i{\text{ is selected}}|{\text{applicant }}i{\text{ is the best}}\right)\cdot P\left({\text{applicant }}i{\text{ is the best}}\right)\\&=\left\cdot {\frac {1}{n}}\\&=\left\cdot {\frac {1}{n}}\\&={\frac {r-1}{n}}\sum _{i=r}^{n}{\frac {1}{i-1}}.\end{aligned}}}
6350:
are faced with problems where the decision alternatives are encountered sequentially. For example, when trying to decide at which gas station along a highway to stop for gas, people might not search enough before stopping. If true, then they would tend to pay more for gas than if they had searched longer. The same may be true when people search online for airline tickets. Experimental research on problems such as the secretary problem is sometimes referred to as
6556:
2049:(1 followed by a hundred zeroes) or even larger. These slips are turned face down and shuffled over the top of a table. One at a time you turn the slips face up. The aim is to stop turning when you come to the number that you guess to be the largest of the series. You cannot go back and pick a previously turned slip. If you turn over all the slips, then of course you must pick the last one turned.
5154:
corresponding optimal decisions and outcomes if the interviewer had been given full information about the value of each applicant. This full-information problem, in which applicants are drawn independently from a known distribution and the interviewer seeks to maximize the expected value of the applicant selected, was originally solved by Moser (1956), Sakaguchi (1961), and Karlin (1962).
1367:, which also has other applications. Modifications for the secretary problem that can be solved by this algorithm include random availabilities of applicants, more general hypotheses for applicants to be of interest to the decision maker, group interviews for applicants, as well as certain models for a random number of applicants.
6492:
applicants coming in random order. When a candidate arrives, she reveals a set of nonnegative numbers. Each value specifies her qualification for one of the jobs. The administrator not only has to decide whether or not to take the applicant but, if so, also has to assign her permanently to one of the
112:
rankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator gains information sufficient
5153:
as they continue the search process by accumulating a set of past data points that they can use to evaluate new candidates as they arrive. A benefit of this so-called partial-information model is that decisions and outcomes achieved given the relative rank information can be directly compared to the
6349:
of actual people in secretary problem situations. In large part, this work has shown that people tend to stop searching too soon. This may be explained, at least in part, by the cost of evaluating candidates. In real world settings, this might suggest that people do not search enough whenever they
1512:
The essence of the model is based on the idea that life is sequential and that real-world problems pose themselves in real time. Also, it is easier to estimate times in which specific events (arrivals of applicants) should occur more frequently (if they do) than to estimate the distribution of the
4773:
Learning in the partial-information sequential search paradigm. The numbers display the expected values of applicants based on their relative rank (out of m total applicants seen so far) at various points in the search. Expectations are calculated based on the case when their values are uniformly
3923:
Finding the single best applicant might seem like a rather strict objective. One can imagine that the interviewer would rather hire a higher-valued applicant than a lower-valued one, and not only be concerned with getting the best. That is, the interviewer will derive some value from selecting an
2014:
The 1/e-law is sometimes confused with the solution for the classical secretary problem described above because of the similar role of the number 1/e. However, in the 1/e-law, this role is more general. The result is also stronger, since it holds for an unknown number of applicants and since the
6442:, who (together with J. R. Pounder) provided a correct analysis for publication in the magazine. Soon afterwards, several mathematicians wrote to Gardner to tell him about the equivalent problem they had heard via the grapevine, all of which can most likely be traced to Flood's original work.
6437:
in
Scientific American, February 1960. He had heard about it from John H. Fox Jr., and L. Gerald Marnie, who had independently come up with an equivalent problem in 1958; they called it the "game of googol". Fox and Marnie did not know the optimum solution; Gardner asked for advice from
5148:
A more general form of this problem introduced by Palley and Kremer (2014) assumes that as each new applicant arrives, the interviewer observes their rank relative to all of the applicants that have been observed previously. This model is consistent with the notion of an interviewer
2858:
By the rules of the game, Alice's sequence must be exchangeable, but to do well in the game, Alice should not pick it to be independent. If Alice samples the numbers independently from some fixed distribution, it would allow Bob to do better. To see this intuitively, imagine if
4626:
1572:
of rankable applicants. The goal is to maximize the probability of selecting only the best under the hypothesis that all arrival orders of different ranks are equally likely. Suppose that all applicants have the same, but independent to each other, arrival time density
325:
is used to mean "reject immediately after the interview". Since the objective in the problem is to select the single best applicant, only candidates will be considered for acceptance. The "candidate" in this context corresponds to the concept of record in permutation.
274:. One reason why the secretary problem has received so much attention is that the optimal policy for the problem (the stopping rule) is simple and selects the single best candidate about 37% of the time, irrespective of whether there are 100 or 100 million applicants.
3243:
2853:
6366:
research on information integration, or the representation of belief, in perceptual decision-making tasks using both animal and human subjects, there is relatively little known about how the decision to stop gathering information is arrived at.
354:. It can be shown that the optimal strategy lies in this class of strategies. (Note that we should never choose an applicant who is not the best we have seen so far, since they cannot be the best overall applicant.) For an arbitrary cutoff
312:
The objective of the general solution is to have the highest probability of selecting the best applicant of the whole group. This is the same as maximizing the expected payoff, with payoff defined to be one for the best applicant and zero
3974:
is given by the true value of the selected applicant. For example, if he/she selects an applicant whose true value is 0.8, then he/she will earn 0.8. The interviewer's objective is to maximize the expected value of the selected applicant.
5008:
the interviewer will begin accepting applicants sooner in the cardinal payoff version than in the classical version where the objective is to select the single best applicant. Note that this is not an asymptotic result: It holds for all
1375:
The solution of the secretary problem is only meaningful if it is justified to assume that the applicants have no knowledge of the decision strategy employed, because early applicants have no chance at all and may not show up otherwise.
6271:
4764:
197:
applicants that are interviewed and then stopping at the first applicant who is better than every applicant interviewed so far (or continuing to the last applicant if this never occurs). Sometimes this strategy is called the
1483:
but typically lower. This can be understood in the context of having a "price" to pay for not knowing the number of applicants. However, in this model the price is high. Depending on the choice of the distribution of
6143:
4170:
4069:
5300:
3892:-th encountered candidate. Note, that this rule does not necessarily skip any applicants; it only considers how many candidates have been observed, not how deep the decision maker is in the applicant sequence.
6001:
1067:
3846:, the answer is yes: Alice can choose random numbers (which are dependent random variables) in such a way that Bob cannot play better than using the classical stopping strategy based on the relative ranks.
3092:
3876:
applicants; thereafter, select the first encountered candidate (i.e., an applicant with relative rank 1). This rule has as a special case the optimal policy for the classical secretary problem for which
2702:
4341:
4832:
6327:
369:
5637:
5228:
4956:
4227:
3962:
on . Similar to the classical problem described above, the interviewer only observes whether each applicant is the best so far (a candidate), must accept or reject each on the spot, and
6080:
4986:
4257:
2045:
Ask someone to take as many slips of paper as he pleases, and on each slip write a different positive number. The numbers may range from small fractions of 1 to a number the size of a
6493:
jobs. The objective is to find an assignment where the sum of qualifications is as big as possible. This problem is identical to finding a maximum-weight matching in an edge-weighted
1711:
5578:
7391:
Kesselheim, Thomas; Radke, Klaus; Tönnis, Andreas; Vöcking, Berthold (2013). "An
Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions".
1477:
5869:
2429:
2364:
2272:
2181:
6426:
and J. Robbins, outlining a proof of the optimum strategy, with an appendix by R. Palermo who proved that all strategies are dominated by a strategy of the form "reject the first
5816:
5479:
3868:
derived the expected success probabilities for several psychologically plausible heuristics that might be employed in the secretary problem. The heuristics they examined were:
1948:, whereas this value 1/e was now achieved as a lower bound for the success probability, and this in a model with arguably much weaker hypotheses (see e.g. Math. Reviews 85:m).
1353:
5398:
5143:
3063:
2467:
1744:
3741:
6524:
By a generalization of the classic algorithm for the secretary problem, it is possible to obtain an assignment where the expected sum of qualifications is only a factor of
4902:
4333:
3815:
2697:
2647:
2597:
1814:
5366:
4281:
3781:
3004:
3493:
3343:
4774:
distributed between 0 and 1. Relative rank information allows the interviewer to more finely evaluate applicants as they accumulate more data points to compare them to.
1479:(Presman and Sonin, 1972). For this model, the optimal solution is in general much harder, however. Moreover, the optimal success probability is now no longer around 1/
5174:
calls this the "postdoc" problem arguing that the "best" will go to
Harvard. For this problem, the probability of success for an even number of applicants is exactly
4665:
195:
2543:
2505:
3844:
2918:
7794:
7595:
Flood, Merrill R. (1958). "Proof of the optimum strategy". Letter to Martin
Gardner. Martin Gardner papers series 1, box 5, folder 19: Stanford University Archives.
6707:
5896:
5787:
5436:
3712:
3589:
3397:
3370:
3087:
2295:
2207:
2009:
1989:
1854:
1834:
1768:
121:
of tracking the running maximum (and who achieved it), and selecting the overall maximum at the end. The difficulty is that the decision must be made immediately.
113:
to rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants. The question is about the optimal strategy (
6172:
5930:
3278:
2883:
2088:
Bob, the stopping player, observes the actual values and can stop turning cards whenever he wants, winning if the last card turned has the overall maximal number.
1504:, the optimal win probability can approach zero. Looking for ways to cope with this new problem led to a new model yielding the so-called 1/e-law of best choice.
252:
224:
154:
5722:
application officers, each holding one letter. You keep interviewing the candidates and rank them on a chart that every application officer can see. Now officer
2964:
2941:
7487:
Bearden, J. Neil; Rapoport, Amnon; Murphy, Ryan O. (September 2006). "Sequential
Observation and Selection with Rank-Dependent Payoffs: An Experimental Study".
6542:
6515:
6490:
6177:
6021:
5760:
5740:
5720:
5700:
5680:
5660:
5507:
5340:
5320:
5107:
5087:
5067:
5047:
5027:
5006:
4926:
4872:
4852:
4301:
4193:
3945:
3612:
3310:
2083:
1969:
1946:
1913:
1888:
1643:
1591:
1570:
1502:
1417:
1397:
1284:
1231:
1178:
1098:
941:
272:
110:
1623:
1550:
6414:, who called it the fiancée problem in a lecture he gave that year. He referred to it several times during the 1950s, for example, in a conference talk at
4677:
5509:
choices and wins if any choice is the best. An optimal strategy for this problem belongs to the class of strategies defined by a set of threshold numbers
6378:(MDP) was used to quantify the value of continuing to search versus committing to the current option. Decisions to take versus decline an option engaged
2366:, and not on their numerical values. In other words, it is as if someone secretly intervened after Alice picked her numbers, and changed each number in
1951:
However, there are many other strategies that achieve (i) and (ii) and, moreover, perform strictly better than the 1/e-strategy simultaneously for all
1399:
must be known in advance, which is rarely the case. One way to overcome this problem is to suppose that the number of applicants is a random variable
117:) to maximize the probability of selecting the best applicant. If the decision can be deferred to the end, this can be solved by the simple maximum
7805:
Ketelaar, Timothy; Todd, Peter M. (2001). "Framing Our
Thoughts: Ecological Rationality as Evolutionary Psychology's Answer to the Frame Problem".
1991:
provided that at least one applicant arrived before this time, and otherwise selects (if possible) the second relatively best candidate after time
8457:
2091:
Bob wants to guess the maximal number with the highest possible probability, while Alice's goal is to keep this probability as low as possible.
7921:
Seale, D.A.; Rapoport, A. (1997). "Sequential decision making with relative ranks: An experimental investigation of the 'secretary problem'".
6418:
on 9 May 1958, and it eventually became widely known in the folklore although nothing was published at the time. In 1958 he sent a letter to
2699:, the optimal relative rank stopping strategy is the optimal stopping rule for the secretary problem, given above, with a winning probability
7251:
Heekeren, Hauke R.; Marrett, Sean; Ungerleider, Leslie G. (9 May 2008). "The neural systems that mediate human perceptual decision making".
6085:
4077:
3989:
5244:
2115:
Alice first writes down n numbers, which are then shuffled. So, their ordering does not matter, meaning that Alice's numbers must be an
5935:
1928:, came as a surprise. The reason was that a value of about 1/e had been considered before as being out of reach in a model for unknown
977:
5230:. This probability tends to 1/4 as n tends to infinity illustrating the fact that it is easier to pick the best than the second-best.
6464:
long before that, who spent 2 years investigating 11 candidates for marriage during 1611 -- 1613 after the death of his first wife.
5145:, with the same convexity claims as before. For other known distributions, optimal play can be calculated via dynamic programming.
7448:
Bearden, J.N.; Murphy, R.O.; Rapoport, A. (2005). "A multi-attribute extension of the secretary problem: Theory and experiments".
4621:{\displaystyle V_{n}(c)=\sum _{t=c}^{n-1}\left\left({\frac {1}{t+1}}\right)+\left{\frac {1}{2}}={\frac {2cn-{c}^{2}+c-n}{2cn}}.}
166:), and that the latter holds even in a much greater generality. The optimal stopping rule prescribes always rejecting the first
3280:, if Bob plays the optimal relative-rank stoppings strategy, then Bob has winning probability 1/2. Surprisingly, Alice has no
7882:
7822:
7679:
7408:
6795:
5789:. (Unsent letters of acceptance are by default given to the last applicants, the same as in the standard secretary problem.)
4781:
1379:
One important drawback for applications of the solution of the classical secretary problem is that the number of applicants
1363:
This problem and several modifications can be solved (including the proof of optimality) in a straightforward manner by the
6518:
6282:
309:
The decision to accept or reject an applicant can be based only on the relative ranks of the applicants interviewed so far.
3399:. Now, Bob can win with probability strictly greater than 1/2. Suppose Alice's numbers are different, then conditional on
306:
Immediately after an interview, the interviewed applicant is either accepted or rejected, and the decision is irrevocable.
8046:
5583:
7965:
7834:
3238:{\displaystyle Pr(X_{\tau }=\max _{i\in 1:n}X_{i})\leq \max _{r\in 1:n}{\frac {r-1}{n}}\sum _{i=r}^{n}{\frac {1}{i-1}}}
7075:
3959:
92:
The basic form of the problem is the following: imagine an administrator who wants to hire the best secretary out of
6456:
Ferguson has an extensive bibliography and points out that a similar (but different) problem had been considered by
8442:
3966:
accept the last one if he/she is reached. (To be clear, the interviewer does not learn the actual relative rank of
2848:{\displaystyle Pr(X_{\tau }=\max _{i\in 1:n}X_{i})=\max _{r\in 1:n}{\frac {r-1}{n}}\sum _{i=r}^{n}{\frac {1}{i-1}}}
1971:>2. A simple example is the strategy which selects (if possible) the first relatively best candidate after time
8021:
sequence A054404 (Number of daughters to wait before picking in sultan's dowry problem with n daughters)
7204:"Response of Neurons in the Lateral Intraparietal Area during a Combined Visual Discrimination Reaction Time Task"
7037:
4175:
As in the classical problem, the optimal policy is given by a threshold, which for this problem we will denote by
5177:
4931:
4202:
3924:
applicant that is not necessarily the best, and the derived value increases with the value of the one selected.
6472:
The secretary problem can be generalized to the case where there are multiple different jobs. Again, there are
6383:
6351:
6026:
4961:
4232:
2100:
3907:. The figure (shown on right) displays the expected success probabilities for each heuristic as a function of
1651:
8452:
7942:
Stein, W.E.; Seale, D.A.; Rapoport, A. (2003). "Analysis of heuristic solutions to the best choice problem".
5512:
2116:
226:
stopping rule, because the probability of stopping at the best applicant with this strategy is already about
7652:
Girdhar, Yogesh; Dudek, Gregory (2009). "Optimal Online Data
Sampling or How to Hire the Best Secretaries".
1422:
321:
is defined as an applicant who, when interviewed, is better than all the applicants interviewed previously.
7606:
5821:
2369:
2304:
2212:
2121:
5795:
350: − 1 applicants), and then selects the first subsequent applicant that is better than applicant
8437:
7600:
7296:"Frontal-Parietal and Limbic-Striatal Activity Underlies Information Sampling in the Best Choice Problem"
5441:
8447:
6370:
Researchers have studied the neural bases of solving the secretary problem in healthy volunteers using
3970:
applicant. He/she learns only whether the applicant has relative rank 1.) However, in this version the
2943:, then he can quite confidently turn over the second number, and if Bob turns over one number and sees
5302:
candidates without picking any one of them, then pick every candidate that is better than those first
3854:
The remainder of the article deals again with the secretary problem for a known number of applicants.
1324:
7990:
5371:
5112:
3016:
2434:
2032:
1716:
158:
8002:
7462:
7130:
Bearden, Murphy, and
Rapoport, 2006; Bearden, Rapoport, and Murphy, 2006; Seale and Rapoport, 1997;
3720:
1321:
The probability of selecting the best applicant in the classical secretary problem converges toward
8432:
4877:
4306:
3786:
2652:
2602:
2552:
1773:
1513:
number of specific events which will occur. This idea led to the following approach, the so-called
7662:
5345:
303:
The applicants are interviewed sequentially in random order, with each order being equally likely.
7786:
6608:
6461:
6375:
6338:
5162:
There are several variants of the secretary problem that also have simple and elegant solutions.
4262:
3746:
2969:
2855:
Alice's goal then is to make sure Bob cannot do better than the relative-rank stopping strategy.
2106:
Bob observes the actual values written on the cards, which he can use in his decision procedures.
7427:
Bearden, J.N. (2006). "A new secretary problem with rank-based selection and cardinal payoffs".
7131:
6857:
5742:
would send their letter of acceptance to the first candidate that is better than all candidates
3402:
3315:
7997:
7657:
7457:
6342:
4634:
2099:
Alice does not have to write numbers uniformly at random. She may write them according to any
169:
7892:
Sardelis, Dimitris A.; Valahas, Theodoros M. (March 1999). "Decision Making: A Golden Rule".
7069:
3895:
Successive non-candidate rule (SNCR): Select the first encountered candidate after observing
3289:
2510:
2472:
2183:. Alice's strategy is then just picking the trickiest exchangeable random variable sequence.
911:
is the best applicant, then it is selected if and only if the best applicant among the first
5238:
Consider the problem of picking the k best secretaries out of n candidates, using k tries.
3823:
2888:
1920:(iii) selects, if there is at least one applicant, none at all with probability exactly 1/e.
7156:
5874:
5765:
5407:
3621:
3498:
3375:
3348:
3072:
2280:
2192:
1994:
1974:
1839:
1819:
1753:
899:= 1, but in this case the only feasible policy is to select the first applicant, and hence
7832:
Matsui, T; Ano, K (2016). "Lower bounds for Bruss' odds problem with multiple stoppings".
6266:{\displaystyle e^{-1}+e^{-{\frac {3}{2}}}+e^{-{\frac {47}{24}}}+e^{-{\frac {2761}{1152}}}}
5170:
One variant replaces the desire to pick the best with the desire to pick the second-best.
8:
8030:
7986:
7570:"A Unified Approach to a Class of Best Choice Problems with an Unknown Number of Options"
7033:
6598:
6151:
5909:
5171:
3257:
2862:
2037:
1140:
229:
201:
131:
118:
54:
8057:
7160:
6402:
representation encode threshold crossings that trigger decisions to commit to a choice.
2946:
2923:
7909:
7861:
7843:
7774:
7754:
7714:
7685:
7631:
7475:
7320:
7295:
7276:
7228:
7219:
7203:
6980:
6697:
6672:
6583:
6575:
6527:
6500:
6475:
6395:
6006:
5745:
5725:
5705:
5685:
5665:
5645:
5492:
5325:
5305:
5092:
5072:
5052:
5032:
5012:
4991:
4911:
4857:
4837:
4286:
4178:
3930:
3597:
3295:
2068:
2024:
1954:
1931:
1898:
1873:
1628:
1576:
1555:
1487:
1402:
1382:
1269:
1216:
1163:
1083:
926:
257:
95:
7955:
6725:
2966:, then he can quite confidently pick the first number. Alice can do better by picking
1596:
1523:
8027:
7878:
7818:
7778:
7675:
7404:
7325:
7268:
7233:
7184:
7179:
7144:
7057:
7015:
6984:
6972:
6937:
6932:
6915:
6877:
6858:"Sequential Search and Learning from Rank Feedback: Theory and Experimental Evidence"
6791:
6701:
6689:
6569:
6415:
4759:{\displaystyle {\frac {\partial V}{\partial c}}={\frac {-{c}^{\,2}+n}{2{c}^{\,2}n}}.}
1520:
The model is defined as follows: An applicant must be selected on some time interval
300:
The applicants, if all seen together, can be ranked from best to worst unambiguously.
163:
7963:
Vanderbei, R. J. (November 1980). "The
Optimal Choice of a Subset of a Population".
7865:
6955:
Rose, John S. (1982). "Selection of nonextremal candidates from a random sequence".
6842:
7974:
7951:
7930:
7901:
7853:
7810:
7766:
7739:
7706:
7689:
7667:
7623:
7581:
7554:
7524:
7496:
7479:
7467:
7436:
7396:
7315:
7307:
7280:
7260:
7223:
7215:
7174:
7164:
7049:
7007:
6964:
6927:
6869:
6783:
6681:
6667:
6647:
6593:
6561:
6517:
nodes of one side arrive online in random order. Thus, it is a special case of the
6411:
6387:
3978:
Since the applicant's values are i.i.d. draws from a uniform distribution on , the
50:
4195:, at which the interviewer should begin accepting candidates. Bearden showed that
2015:
model based on an arrival time distribution F is more tractable for applications.
7538:
7508:
7400:
6787:
6494:
6450:
6419:
6379:
6346:
3948:
3285:
1925:
62:
7814:
3857:
282:
Although there are many variations, the basic problem can be stated as follows:
6775:
6588:
6434:
6391:
6371:
3979:
2028:
1364:
125:
8050:
7770:
7471:
7440:
8426:
7744:
7727:
7586:
7569:
7559:
7542:
7529:
7512:
7061:
7019:
6976:
6941:
6881:
6693:
6603:
6457:
6423:
6399:
6398:. Therefore, brain regions previously implicated in evidence integration and
2187:
2058:
335:
114:
7755:"The best choice problem with random arrivals: How to beat the 1/e-strategy"
7614:
Freeman, P.R. (1983). "The secretary problem and its extensions: A review".
7311:
7011:
6652:
6635:
7934:
7857:
7500:
7329:
7272:
7237:
7053:
6873:
6363:
21:
7697:
Gilbert, J; Mosteller, F (1966). "Recognizing the
Maximum of a Sequence".
7188:
7169:
6998:
Szajowski, Krzysztof (1982). "Optimal choice of an object with ath rank".
7978:
6726:"Mathematicians suggest the "37% rule" for your life's biggest decisions"
639:
25:
Graphs of probabilities of getting the best candidate (red circles) from
7671:
6685:
4769:
7913:
7875:
The mating mind: how sexual choice shaped the evolution of human nature
7718:
7635:
7395:. Lecture Notes in Computer Science. Vol. 8125. pp. 589–600.
6968:
2027:, 1989), it's claimed the secretary problem first appeared in print in
58:
7616:
International Statistical Review / Revue Internationale de Statistique
3899:
non-candidates (i.e., applicants with relative rank > 1).
1895:(ii) is a minimax-optimal strategy for the selector who does not know
8035:
7447:
6439:
1645:
denote the corresponding arrival time distribution function, that is
7905:
7710:
7627:
7264:
6138:{\displaystyle p_{i}=\lim _{n\rightarrow \infty }{\frac {a_{i}}{n}}}
1856:
which is better than all preceding ones. Then this strategy, called
1816:
Consider the strategy to wait and observe all applicants up to time
8025:
4165:{\displaystyle E_{t}=E\left(X_{t}|I_{t}=1\right)={\frac {t}{t+1}}.}
4064:{\displaystyle x_{t}=\max \left\{x_{1},x_{2},\ldots ,x_{t}\right\}}
7848:
5295:{\displaystyle r=\left\lfloor {\frac {n}{ke^{1/k}}}\right\rfloor }
8017:
5996:{\displaystyle e^{-1}+e^{-{\frac {3}{2}}},(n\rightarrow \infty )}
3292:. Concretely, Bob can play this strategy: sample a random number
3281:
2885:, and Alice is to pick both numbers from the normal distribution
1124:
increases, and the best applicant is selected with probability 1/
1062:{\displaystyle P(x)=x\int _{x}^{1}{\frac {1}{t}}\,dt=-x\ln(x)\;.}
128:. It implies that the optimal win probability is always at least
7809:. Studies in Cognitive Systems. Vol. 27. pp. 179–211.
6774:
Cover, Thomas M. (1987), Cover, Thomas M.; Gopinath, B. (eds.),
6430:
unconditionally, then accept the next candidate who is better".
2065:
Alice, the informed player, writes secretly distinct numbers on
1836:
and then to select, if possible, the first candidate after time
7099:
7097:
6555:
3955:
3013:
Does there exist an exchangeable sequence of random variables
2431:
into its relative rank (breaking ties randomly). For example,
4303:
applicants, the expected payoff for some arbitrary threshold
8261:# Define a function to print the results as a Markdown table
8105:# Define the function for which you want to find the maximum
7390:
7348:
7094:
5241:
In general, the optimal decision method starts by observing
2920:, independently. Then if Bob turns over one number and sees
124:
The shortest rigorous proof known so far is provided by the
8020:
7543:"A note on bounds for the odds theorem of optimal stopping"
6410:
The secretary problem was apparently introduced in 1949 by
636:
7250:
7202:
Roitman, Jamie D.; Shadlen, Michael N. (1 November 2002).
2649:
is just the uniform distribution over all permutations on
8195:# Define a function to solve the problem for a specific n
2599:. Now, since the only exchangeable random permutation on
2095:
The difference with the basic secretary problem are two:
358:, the probability that the best applicant is selected is
7793:, Vol. 97, 126-133 (2009). (For French translation, see
7114:
7112:
7082:
6916:"Dynamic programming of some sequential sampling design"
4283:.) This follows from the fact that given a problem with
7920:
7044:. Annales Societatis Mathematicae Polonae, Series III.
7002:. Annales Societatis Mathematicae Polonae, Series III.
5322:
candidates until we run out of candidates or picks. If
5484:
5165:
4928:, the optimal integer-valued threshold must be either
4827:{\displaystyle \partial ^{\,2}V/\partial c^{\,2}<0}
919: − 1 applicants that were rejected. Letting
16:
Mathematical problem involving optimal stopping theory
7654:
2009 Canadian Conference on Computer and Robot Vision
7486:
7109:
6530:
6503:
6478:
6322:{\displaystyle p_{5}=e^{-{\frac {4162637}{1474560}}}}
6285:
6180:
6154:
6088:
6029:
6009:
5938:
5912:
5877:
5824:
5798:
5768:
5748:
5728:
5708:
5688:
5668:
5648:
5586:
5515:
5495:
5444:
5410:
5374:
5348:
5328:
5308:
5247:
5233:
5180:
5115:
5095:
5075:
5055:
5035:
5015:
4994:
4964:
4934:
4914:
4880:
4860:
4840:
4784:
4680:
4637:
4344:
4309:
4289:
4265:
4235:
4205:
4181:
4080:
3992:
3933:
3872:
The cutoff rule (CR): Do not accept any of the first
3826:
3789:
3749:
3723:
3624:
3600:
3501:
3405:
3378:
3351:
3318:
3298:
3260:
3095:
3075:
3019:
2972:
2949:
2926:
2891:
2865:
2705:
2655:
2605:
2555:
2513:
2475:
2437:
2372:
2307:
2283:
2215:
2195:
2124:
2071:
1997:
1977:
1957:
1934:
1901:
1876:
1842:
1822:
1776:
1756:
1719:
1654:
1631:
1599:
1579:
1558:
1526:
1490:
1425:
1405:
1385:
1327:
1272:
1219:
1166:
1086:
980:
929:
367:
260:
232:
204:
172:
134:
98:
7923:
Organizational Behavior and Human Decision Processes
7645:
New Mathematical Diversions from Scientific American
7372:
6551:
3495:, Bob wins with probability 1/2, but conditional on
2023:
In the article "Who solved the Secretary problem?" (
53:
theory that is studied extensively in the fields of
7872:
3861:
Expected success probabilities for three heuristics
3284:strategy, which is closely related to a paradox of
2549:Alice played an exchangeable random permutation on
907:. This sum is obtained by noting that if applicant
7941:
7143:Shadlen, M. N.; Newsome, W. T. (23 January 1996).
6822:
6755:
6536:
6509:
6484:
6321:
6265:
6166:
6137:
6074:
6015:
5995:
5924:
5890:
5863:
5810:
5781:
5754:
5734:
5714:
5694:
5674:
5654:
5632:{\displaystyle a_{1}>a_{2}>\cdots >a_{r}}
5631:
5572:
5501:
5473:
5430:
5392:
5360:
5334:
5314:
5294:
5222:
5137:
5101:
5081:
5061:
5041:
5021:
5000:
4980:
4950:
4920:
4896:
4866:
4846:
4826:
4758:
4659:
4620:
4327:
4295:
4275:
4251:
4221:
4187:
4164:
4063:
3939:
3865:
3838:
3809:
3775:
3735:
3706:
3606:
3583:
3487:
3391:
3364:
3337:
3304:
3272:
3237:
3081:
3057:
2998:
2958:
2935:
2912:
2877:
2847:
2691:
2641:
2591:
2537:
2499:
2461:
2423:
2358:
2289:
2266:
2201:
2175:
2077:
2003:
1983:
1963:
1940:
1907:
1882:
1848:
1828:
1808:
1762:
1738:
1705:
1637:
1617:
1585:
1564:
1544:
1496:
1471:
1411:
1391:
1347:
1278:
1225:
1172:
1147:and probability of selecting the best alternative
1092:
1061:
935:
884:
266:
246:
218:
189:
148:
104:
7360:
7336:
7294:Costa, V. D.; Averbeck, B. B. (18 October 2013).
6920:Journal of Mathematical Analysis and Applications
6810:
6743:
8424:
7807:Conceptual Challenges in Evolutionary Psychology
7696:
7103:
6103:
4006:
3669:
3634:
3546:
3511:
3450:
3415:
3157:
3119:
2767:
2729:
7891:
7699:Journal of the American Statistical Association
7149:Proceedings of the National Academy of Sciences
6841:Jones, Maxwell; Nene, Advait (20 August 2024).
6467:
5368:, then the probability of success converges to
3783:such that Bob's winning probability is at most
3743:, Alice can construct an exchangeable sequence
7201:
7142:
7038:"The postdoc variant of the secretary problem"
6780:Open Problems in Communication and Computation
971:, the sum can be approximated by the integral
338:. Under it, the interviewer rejects the first
329:
293:applicants for the position, and the value of
8372:# Convert the DataFrame to Markdown and print
7613:
7293:
6895:Moser, Leo (1956). "On a problem of Cayley".
6856:Palley, Asa B.; Kremer, Mirko (8 July 2014).
5049:secretaries has a fixed, distinct value from
2301:iff it depends on only the relative ranks of
2061:with two antagonistic players. In this game:
915: − 1 applicants is among the first
7991:The postdoc variant of the secretary problem
7804:
7651:
7088:
6855:
4975:
4965:
4945:
4935:
4246:
4236:
4216:
4206:
2686:
2656:
2636:
2606:
2586:
2556:
7759:Stochastic Processes and Their Applications
6544:less than an optimal (offline) assignment.
6422:, with copies to a dozen friends including
5223:{\displaystyle {\frac {0.25n^{2}}{n(n-1)}}}
4951:{\displaystyle \lfloor {\sqrt {n}}\rfloor }
4222:{\displaystyle \lfloor {\sqrt {n}}\rfloor }
6948:
6629:
6627:
6625:
6623:
6023:, the probability of winning converges to
5932:, the probability of winning converges to
3918:
3009:So the fully formal statement is as below:
1055:
8001:
7985:
7962:
7847:
7831:
7743:
7661:
7585:
7558:
7528:
7461:
7319:
7227:
7178:
7168:
7118:
7032:
6997:
6931:
6913:
6840:
6651:
6276:
6075:{\displaystyle p_{1}+p_{2}+\cdots +p_{r}}
5901:
5401:
4981:{\displaystyle \lceil {\sqrt {n}}\rceil }
4812:
4790:
4741:
4717:
4252:{\displaystyle \lceil {\sqrt {n}}\rceil }
1720:
1507:
1024:
342: − 1 applicants (let applicant
8044:
7944:European Journal of Operational Research
7378:
7145:"Motion perception: seeing and deciding"
6633:
6433:The first publication was apparently by
6003:. More generally, for positive integers
4768:
3927:To model this problem, suppose that the
3856:
3849:
2053:Ferguson pointed out that the secretary
1706:{\displaystyle F(t)=\int _{0}^{t}f(s)ds}
334:The optimal policy for the problem is a
20:
7642:
7426:
7354:
7026:
6828:
6782:, New York, NY: Springer, p. 152,
6761:
6723:
6660:
6620:
6332:
6279:gave a general algorithm. For example,
5573:{\displaystyle (a_{1},a_{2},...,a_{r})}
3947:applicants have "true" values that are
3888:Candidate count rule (CCR): Select the
1358:
8425:
7752:
7725:
6991:
6816:
6749:
5157:
3903:Each heuristic has a single parameter
2545:with equal probability. This makes it
1890:a success probability of at least 1/e,
1472:{\displaystyle P(N=k)_{k=1,2,\cdots }}
8458:Mathematical optimization in business
8047:"Optimal Search (Secretary Problems)"
8026:
7594:
7567:
7537:
7507:
7366:
7342:
6894:
6843:"Secretary Problem Variant Deep Dive"
6773:
6362:While there is a substantial body of
5864:{\displaystyle a_{i}\sim ne^{-k_{i}}}
5489:In this variant, a player is allowed
5438:, then the probability of success is
2424:{\displaystyle X_{1},X_{2},...,X_{n}}
2359:{\displaystyle X_{1},X_{2},...,X_{n}}
2267:{\displaystyle X_{1},X_{2},...,X_{n}}
2176:{\displaystyle X_{1},X_{2},...,X_{n}}
2117:exchangeable random variable sequence
2110:
2018:
1112:. Thus, the optimal cutoff tends to
8402:# Print the table for n from 1 to 10
6954:
6666:
6357:
5811:{\displaystyle n\rightarrow \infty }
5662:letters of acceptance labelled from
5642:Specifically, imagine that you have
4259:. (In fact, whichever is closest to
2186:Bob's strategy is formalizable as a
85:. Its solution is also known as the
6636:"Who Solved the Secretary Problem?"
6634:Ferguson, Thomas S. (August 1989).
5485:Pick the best, using multiple tries
5166:Pick the second-best, using one try
1100:, setting it to 0, and solving for
286:There is a single position to fill.
13:
7966:Mathematics of Operations Research
7835:Mathematics of Operations Research
7728:"A solution to the game of Googol"
7450:Journal of Mathematical Psychology
7429:Journal of Mathematical Psychology
7220:10.1523/JNEUROSCI.22-21-09475.2002
6113:
5987:
5805:
5474:{\displaystyle {\frac {1}{n/2+1}}}
5355:
5234:Pick the top-k ones, using k tries
4804:
4786:
4692:
4684:
1155:are shown in the following table.
346:be the best applicant among these
49:demonstrates a scenario involving
14:
8469:
8059:Optimal Stopping and Applications
8011:
7894:The American Mathematical Monthly
6914:Sakaguchi, Minoru (1 June 1961).
1139:can also be obtained by standard
7568:Bruss, F. Thomas (August 1984).
6724:Thomson, Jonny (21 April 2022).
6670:(2009). "Knowing When to Stop".
6568:
6554:
5029:. Interestingly, if each of the
3866:Stein, Seale & Rapoport 2003
3618:random distribution, as long as
3591:, Bob wins with probability 1.
3006:that are positively correlated.
1860:, has the following properties:
1348:{\displaystyle 1/e\approx 0.368}
1143:methods. The optimal thresholds
7384:
7287:
7244:
7195:
7136:
7124:
6907:
6888:
6849:
6834:
5393:{\displaystyle {\frac {1}{ek}}}
5138:{\displaystyle c={\sqrt {n}}-1}
3058:{\displaystyle X_{1},...,X_{n}}
2462:{\displaystyle 0.2,0.3,0.3,0.1}
2299:relative rank stopping strategy
1924:The 1/e-law, proved in 1984 by
1739:{\displaystyle \,0\leq t\leq T}
8072:
7513:"Sum the odds to one and stop"
6767:
6717:
6449:-law of best choice is due to
6352:behavioral operations research
6110:
5990:
5984:
5978:
5802:
5567:
5516:
5352:
5214:
5202:
4834:for all permissible values of
4654:
4648:
4361:
4355:
4113:
3736:{\displaystyle \epsilon >0}
3701:
3698:
3672:
3663:
3637:
3631:
3578:
3575:
3549:
3540:
3514:
3508:
3482:
3479:
3453:
3444:
3418:
3412:
3150:
3102:
2907:
2895:
2760:
2712:
2101:joint probability distribution
1786:
1780:
1694:
1688:
1664:
1658:
1612:
1600:
1539:
1527:
1442:
1429:
1419:with a known distribution of
1370:
1052:
1046:
990:
984:
507:
381:
375:
277:
1:
7956:10.1016/S0377-2217(02)00601-X
7643:Gardner, Martin (1966). "3".
7420:
7074:: CS1 maint: date and year (
4897:{\displaystyle c={\sqrt {n}}}
4328:{\displaystyle 1\leq c\leq n}
3810:{\displaystyle 1/2+\epsilon }
2692:{\displaystyle \{1,2,...,n\}}
2642:{\displaystyle \{1,2,...,n\}}
2592:{\displaystyle \{1,2,...,n\}}
1809:{\displaystyle F(\tau )=1/e.}
8279:# Prepare data for the table
7873:Miller, Geoffrey F. (2001).
7401:10.1007/978-3-642-40450-4_50
7104:Gilbert & Mosteller 1966
6933:10.1016/0022-247X(61)90023-3
6788:10.1007/978-1-4612-4808-8_43
6706:For French translation, see
6468:Combinatorial generalization
5361:{\displaystyle n\to \infty }
3594:Note that the random number
2277:We say that a stopping rule
7:
7815:10.1007/978-94-010-0618-7_7
7253:Nature Reviews Neuroscience
7208:The Journal of Neuroscience
6547:
5871:, for some rational number
4988:. Thus, for most values of
4276:{\displaystyle {\sqrt {n}}}
3776:{\displaystyle X_{1},X_{2}}
3249:
2999:{\displaystyle X_{1},X_{2}}
1104:, we find that the optimal
895:The sum is not defined for
644:the best of the first
330:Deriving the optimal policy
10:
8474:
8066:
8062:book by Thomas S. Ferguson
6405:
3488:{\displaystyle Y\not \in }
3338:{\displaystyle X_{1}>Y}
923:tend to infinity, writing
65:. It is also known as the
7771:10.1016/j.spa.2021.12.008
7574:The Annals of Probability
7547:The Annals of Probability
7517:The Annals of Probability
7472:10.1016/j.jmp.2005.08.002
7441:10.1016/j.jmp.2005.11.003
6776:"Pick the Largest Number"
6519:online bipartite matching
3714:has nonzero probability.
2033:Mathematical Games column
1072:Taking the derivative of
8078:
8031:"Sultan's Dowry Problem"
7605:: CS1 maint: location (
7089:Girdhar & Dudek 2009
6614:
4660:{\displaystyle V_{n}(c)}
3986:th applicant given that
2057:remained unsolved, as a
190:{\displaystyle \sim n/e}
8443:Matching (graph theory)
7132:Palley and Kremer, 2014
7012:10.14708/ma.v10i19.1533
6609:Stable marriage problem
6384:dorsolateral prefrontal
6376:Markov decision process
5342:is held constant while
3919:Cardinal payoff variant
2538:{\displaystyle 2,4,3,1}
2500:{\displaystyle 2,3,4,1}
1552:from an unknown number
254:for moderate values of
7935:10.1006/obhd.1997.2683
7858:10.1287/moor.2015.0748
7745:10.1214/aop/1176988613
7587:10.1214/aop/1176993237
7560:10.1214/aop/1068646368
7530:10.1214/aop/1019160340
7501:10.1287/mnsc.1060.0535
7054:10.14708/ma.v49i1.7076
7042:Mathematica Applicanda
6874:10.1287/mnsc.2014.1902
6538:
6511:
6486:
6323:
6267:
6168:
6139:
6076:
6017:
5997:
5926:
5902:Probability of winning
5892:
5865:
5812:
5783:
5756:
5736:
5716:
5696:
5676:
5656:
5633:
5574:
5503:
5475:
5432:
5394:
5362:
5336:
5316:
5296:
5224:
5139:
5103:
5083:
5063:
5043:
5023:
5002:
4982:
4952:
4922:
4898:
4868:
4848:
4828:
4775:
4760:
4661:
4622:
4517:
4425:
4393:
4329:
4297:
4277:
4253:
4223:
4189:
4166:
4065:
3941:
3862:
3840:
3839:{\displaystyle n>2}
3811:
3777:
3737:
3708:
3608:
3585:
3489:
3393:
3366:
3339:
3306:
3274:
3247:
3239:
3216:
3083:
3059:
3000:
2960:
2937:
2914:
2913:{\displaystyle N(0,1)}
2879:
2849:
2826:
2693:
2643:
2593:
2539:
2501:
2463:
2425:
2360:
2291:
2268:
2203:
2177:
2079:
2051:
2005:
1985:
1965:
1942:
1909:
1884:
1850:
1830:
1810:
1764:
1740:
1707:
1639:
1619:
1587:
1566:
1546:
1508:1/e-law of best choice
1498:
1473:
1413:
1393:
1349:
1280:
1227:
1174:
1151:for several values of
1094:
1063:
937:
886:
856:
763:
626:
599:
484:
411:
268:
248:
220:
191:
150:
106:
71:sultan's dowry problem
42:
7797:in the July issue of
7732:Annals of Probability
7647:. Simon and Schuster.
7393:Algorithms – ESA 2013
7312:10.1093/cercor/bht286
7170:10.1073/pnas.93.2.628
7119:Matsui & Ano 2016
6957:J. Optim. Theory Appl
6710:in the July issue of
6653:10.1214/ss/1177012493
6539:
6512:
6487:
6386:cortices, as well as
6324:
6277:Matsui & Ano 2016
6268:
6169:
6140:
6077:
6018:
5998:
5927:
5893:
5891:{\displaystyle k_{i}}
5866:
5813:
5784:
5782:{\displaystyle a_{i}}
5757:
5737:
5717:
5697:
5677:
5657:
5634:
5575:
5504:
5476:
5433:
5431:{\displaystyle k=n/2}
5395:
5363:
5337:
5317:
5297:
5225:
5140:
5104:
5084:
5064:
5044:
5024:
5003:
4983:
4953:
4923:
4899:
4869:
4849:
4829:
4772:
4761:
4662:
4623:
4491:
4399:
4367:
4330:
4298:
4278:
4254:
4224:
4190:
4167:
4066:
3942:
3860:
3850:Heuristic performance
3841:
3812:
3778:
3738:
3709:
3707:{\displaystyle Y\in }
3609:
3586:
3584:{\displaystyle Y\in }
3490:
3394:
3392:{\displaystyle X_{2}}
3367:
3365:{\displaystyle X_{1}}
3340:
3307:
3290:two envelopes paradox
3275:
3240:
3196:
3084:
3082:{\displaystyle \tau }
3060:
3011:
3001:
2961:
2938:
2915:
2880:
2850:
2806:
2694:
2644:
2594:
2540:
2502:
2464:
2426:
2361:
2292:
2290:{\displaystyle \tau }
2269:
2204:
2202:{\displaystyle \tau }
2178:
2080:
2043:
2006:
2004:{\displaystyle \tau }
1986:
1984:{\displaystyle \tau }
1966:
1943:
1910:
1885:
1851:
1849:{\displaystyle \tau }
1831:
1829:{\displaystyle \tau }
1811:
1765:
1763:{\displaystyle \tau }
1741:
1708:
1640:
1620:
1588:
1567:
1547:
1499:
1474:
1414:
1394:
1350:
1281:
1228:
1175:
1095:
1064:
938:
887:
836:
743:
667:is in the first
606:
573:
464:
391:
269:
249:
221:
192:
151:
107:
37:(blue crosses) where
24:
8453:Probability problems
7987:Vanderbei, Robert J.
7979:10.1287/moor.5.4.481
7787:Knowing When to Stop
7656:. pp. 292–298.
7034:Vanderbei, Robert J.
7000:Matematyka Stosowana
6528:
6501:
6476:
6460:in 1875 and even by
6333:Experimental studies
6283:
6178:
6152:
6086:
6027:
6007:
5936:
5910:
5875:
5822:
5796:
5766:
5746:
5726:
5706:
5686:
5666:
5646:
5584:
5513:
5493:
5442:
5408:
5372:
5346:
5326:
5306:
5245:
5178:
5113:
5093:
5073:
5053:
5033:
5013:
4992:
4962:
4932:
4912:
4878:
4858:
4838:
4782:
4678:
4635:
4342:
4307:
4287:
4263:
4233:
4203:
4179:
4078:
3990:
3960:uniform distribution
3931:
3824:
3787:
3747:
3721:
3622:
3614:can be sampled from
3598:
3499:
3403:
3376:
3349:
3316:
3296:
3258:
3093:
3073:
3017:
2970:
2947:
2924:
2889:
2863:
2703:
2653:
2603:
2553:
2511:
2473:
2435:
2370:
2305:
2281:
2213:
2193:
2122:
2069:
1995:
1975:
1955:
1932:
1899:
1874:
1840:
1820:
1774:
1754:
1717:
1652:
1629:
1597:
1577:
1556:
1524:
1488:
1423:
1403:
1383:
1359:Alternative solution
1325:
1270:
1217:
1164:
1131:For small values of
1084:
978:
927:
365:
258:
230:
202:
170:
132:
96:
75:fussy suitor problem
7753:Gnedin, A. (2021).
7726:Gnedin, A. (1994).
7672:10.1109/CRV.2009.30
7161:1996PNAS...93..628S
6686:10.1511/2009.77.126
6640:Statistical Science
6167:{\displaystyle r=4}
5925:{\displaystyle r=2}
5172:Robert J. Vanderbei
5158:Other modifications
3273:{\displaystyle n=2}
2878:{\displaystyle n=2}
2038:Scientific American
1870:(i) yields for all
1684:
1141:dynamic programming
1013:
247:{\displaystyle 1/e}
219:{\displaystyle 1/e}
162:is the base of the
149:{\displaystyle 1/e}
119:selection algorithm
83:best choice problem
55:applied probability
8438:Sequential methods
8053:on 4 January 2017.
8028:Weisstein, Eric W.
7791:American Scientist
7601:cite press release
7489:Management Science
6969:10.1007/BF00934083
6862:Management Science
6673:American Scientist
6584:Assignment problem
6576:Mathematics portal
6534:
6507:
6482:
6396:anterior cingulate
6319:
6263:
6164:
6135:
6117:
6072:
6013:
5993:
5922:
5888:
5861:
5808:
5779:
5752:
5732:
5712:
5692:
5672:
5652:
5629:
5570:
5499:
5471:
5428:
5390:
5358:
5332:
5312:
5292:
5220:
5135:
5099:
5079:
5059:
5039:
5019:
4998:
4978:
4948:
4918:
4894:
4864:
4844:
4824:
4776:
4756:
4657:
4618:
4325:
4293:
4273:
4249:
4219:
4185:
4162:
4061:
3937:
3911:for problems with
3863:
3836:
3807:
3773:
3733:
3704:
3604:
3581:
3485:
3389:
3362:
3335:
3302:
3270:
3235:
3177:
3139:
3079:
3055:
2996:
2959:{\displaystyle +3}
2956:
2936:{\displaystyle -3}
2933:
2910:
2875:
2845:
2787:
2749:
2689:
2639:
2589:
2535:
2497:
2459:
2421:
2356:
2287:
2264:
2199:
2173:
2111:Strategic analysis
2075:
2019:The game of googol
2001:
1981:
1961:
1938:
1905:
1880:
1846:
1826:
1806:
1760:
1736:
1703:
1670:
1635:
1615:
1583:
1562:
1542:
1494:
1469:
1409:
1389:
1345:
1276:
1223:
1170:
1090:
1080:) with respect to
1059:
999:
933:
882:
880:
686:
264:
244:
216:
187:
146:
102:
43:
41:is the sample size
29:applications, and
8448:Optimal decisions
7884:978-0-385-49517-2
7824:978-94-010-3890-4
7681:978-1-4244-4211-9
7410:978-3-642-40449-8
7214:(21): 9475–9489.
6868:(10): 2525–2542.
6797:978-1-4612-4808-8
6668:Hill, Theodore P.
6537:{\displaystyle e}
6510:{\displaystyle n}
6485:{\displaystyle n}
6358:Neural correlates
6347:decision behavior
6345:have studied the
6315:
6259:
6236:
6213:
6133:
6102:
6016:{\displaystyle r}
5971:
5755:{\displaystyle 1}
5735:{\displaystyle i}
5715:{\displaystyle r}
5702:. You would have
5695:{\displaystyle r}
5675:{\displaystyle 1}
5655:{\displaystyle r}
5502:{\displaystyle r}
5469:
5388:
5335:{\displaystyle k}
5315:{\displaystyle r}
5286:
5218:
5127:
5102:{\displaystyle V}
5082:{\displaystyle n}
5062:{\displaystyle 1}
5042:{\displaystyle n}
5022:{\displaystyle n}
5001:{\displaystyle n}
4973:
4943:
4921:{\displaystyle c}
4892:
4867:{\displaystyle V}
4847:{\displaystyle c}
4751:
4699:
4613:
4557:
4538:
4477:
4446:
4296:{\displaystyle n}
4271:
4244:
4214:
4188:{\displaystyle c}
4157:
3940:{\displaystyle n}
3915: = 80.
3717:However, for any
3607:{\displaystyle Y}
3305:{\displaystyle Y}
3233:
3194:
3156:
3118:
2843:
2804:
2766:
2728:
2209:for the sequence
2078:{\displaystyle n}
2031:'s February 1960
1964:{\displaystyle N}
1941:{\displaystyle N}
1908:{\displaystyle N}
1883:{\displaystyle N}
1638:{\displaystyle F}
1586:{\displaystyle f}
1565:{\displaystyle N}
1497:{\displaystyle N}
1412:{\displaystyle N}
1392:{\displaystyle n}
1319:
1318:
1279:{\displaystyle P}
1226:{\displaystyle r}
1173:{\displaystyle n}
1093:{\displaystyle x}
1022:
936:{\displaystyle x}
873:
834:
806:
788:
726:
703:
702: is the best
695:
682:
668:
659:
645:
551:
550: is the best
543:
522:
521: is the best
514:
504:
503: is selected
496:
447:
446: is the best
439:
431:
430: is selected
423:
267:{\displaystyle n}
164:natural logarithm
105:{\displaystyle n}
47:secretary problem
8465:
8418:
8415:
8412:
8409:
8406:
8403:
8400:
8397:
8394:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8370:
8367:
8364:
8361:
8358:
8355:
8352:
8349:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8325:
8322:
8319:
8316:
8313:
8310:
8307:
8304:
8301:
8298:
8295:
8292:
8289:
8286:
8283:
8280:
8277:
8274:
8271:
8268:
8265:
8262:
8259:
8256:
8253:
8250:
8247:
8244:
8241:
8238:
8235:
8232:
8229:
8226:
8223:
8220:
8217:
8214:
8211:
8208:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8178:
8175:
8172:
8169:
8166:
8163:
8160:
8157:
8154:
8151:
8148:
8145:
8142:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8076:
8054:
8049:. Archived from
8041:
8040:
8019:
8007:
8005:
7995:
7982:
7959:
7938:
7917:
7888:
7877:. Anchor Books.
7869:
7851:
7828:
7782:
7749:
7747:
7738:(3): 1588–1595.
7722:
7693:
7665:
7648:
7639:
7610:
7604:
7596:
7591:
7589:
7564:
7562:
7553:(4): 1859–1961.
7541:(October 2003).
7539:Bruss, F. Thomas
7534:
7532:
7523:(3): 1384–1391.
7509:Bruss, F. Thomas
7504:
7495:(9): 1437–1449.
7483:
7465:
7444:
7415:
7414:
7388:
7382:
7376:
7370:
7364:
7358:
7352:
7346:
7340:
7334:
7333:
7323:
7291:
7285:
7284:
7248:
7242:
7241:
7231:
7199:
7193:
7192:
7182:
7172:
7140:
7134:
7128:
7122:
7116:
7107:
7101:
7092:
7086:
7080:
7079:
7073:
7065:
7036:(21 June 2021).
7030:
7024:
7023:
6995:
6989:
6988:
6952:
6946:
6945:
6935:
6911:
6905:
6904:
6892:
6886:
6885:
6853:
6847:
6846:
6838:
6832:
6826:
6820:
6814:
6808:
6807:
6806:
6804:
6771:
6765:
6759:
6753:
6747:
6741:
6740:
6738:
6736:
6721:
6715:
6705:
6664:
6658:
6657:
6655:
6631:
6599:Robbins' problem
6594:Optimal stopping
6578:
6573:
6572:
6564:
6562:Economics portal
6559:
6558:
6543:
6541:
6540:
6535:
6516:
6514:
6513:
6508:
6491:
6489:
6488:
6483:
6412:Merrill M. Flood
6388:ventral striatum
6328:
6326:
6325:
6320:
6318:
6317:
6316:
6308:
6295:
6294:
6272:
6270:
6269:
6264:
6262:
6261:
6260:
6252:
6239:
6238:
6237:
6229:
6216:
6215:
6214:
6206:
6193:
6192:
6173:
6171:
6170:
6165:
6144:
6142:
6141:
6136:
6134:
6129:
6128:
6119:
6116:
6098:
6097:
6081:
6079:
6078:
6073:
6071:
6070:
6052:
6051:
6039:
6038:
6022:
6020:
6019:
6014:
6002:
6000:
5999:
5994:
5974:
5973:
5972:
5964:
5951:
5950:
5931:
5929:
5928:
5923:
5897:
5895:
5894:
5889:
5887:
5886:
5870:
5868:
5867:
5862:
5860:
5859:
5858:
5857:
5834:
5833:
5817:
5815:
5814:
5809:
5788:
5786:
5785:
5780:
5778:
5777:
5761:
5759:
5758:
5753:
5741:
5739:
5738:
5733:
5721:
5719:
5718:
5713:
5701:
5699:
5698:
5693:
5681:
5679:
5678:
5673:
5661:
5659:
5658:
5653:
5638:
5636:
5635:
5630:
5628:
5627:
5609:
5608:
5596:
5595:
5579:
5577:
5576:
5571:
5566:
5565:
5541:
5540:
5528:
5527:
5508:
5506:
5505:
5500:
5480:
5478:
5477:
5472:
5470:
5468:
5458:
5446:
5437:
5435:
5434:
5429:
5424:
5399:
5397:
5396:
5391:
5389:
5387:
5376:
5367:
5365:
5364:
5359:
5341:
5339:
5338:
5333:
5321:
5319:
5318:
5313:
5301:
5299:
5298:
5293:
5291:
5287:
5285:
5284:
5283:
5279:
5259:
5229:
5227:
5226:
5221:
5219:
5217:
5197:
5196:
5195:
5182:
5144:
5142:
5141:
5136:
5128:
5123:
5109:is maximized at
5108:
5106:
5105:
5100:
5088:
5086:
5085:
5080:
5068:
5066:
5065:
5060:
5048:
5046:
5045:
5040:
5028:
5026:
5025:
5020:
5007:
5005:
5004:
4999:
4987:
4985:
4984:
4979:
4974:
4969:
4957:
4955:
4954:
4949:
4944:
4939:
4927:
4925:
4924:
4919:
4903:
4901:
4900:
4895:
4893:
4888:
4874:is maximized at
4873:
4871:
4870:
4865:
4853:
4851:
4850:
4845:
4833:
4831:
4830:
4825:
4817:
4816:
4803:
4795:
4794:
4765:
4763:
4762:
4757:
4752:
4750:
4746:
4745:
4739:
4729:
4722:
4721:
4715:
4705:
4700:
4698:
4690:
4682:
4667:with respect to
4666:
4664:
4663:
4658:
4647:
4646:
4631:Differentiating
4627:
4625:
4624:
4619:
4614:
4612:
4601:
4588:
4587:
4582:
4563:
4558:
4550:
4548:
4544:
4543:
4539:
4534:
4523:
4516:
4505:
4482:
4478:
4476:
4462:
4456:
4452:
4451:
4447:
4442:
4431:
4424:
4413:
4392:
4381:
4354:
4353:
4334:
4332:
4331:
4326:
4302:
4300:
4299:
4294:
4282:
4280:
4279:
4274:
4272:
4267:
4258:
4256:
4255:
4250:
4245:
4240:
4228:
4226:
4225:
4220:
4215:
4210:
4194:
4192:
4191:
4186:
4171:
4169:
4168:
4163:
4158:
4156:
4142:
4137:
4133:
4126:
4125:
4116:
4111:
4110:
4090:
4089:
4070:
4068:
4067:
4062:
4060:
4056:
4055:
4054:
4036:
4035:
4023:
4022:
4002:
4001:
3949:random variables
3946:
3944:
3943:
3938:
3845:
3843:
3842:
3837:
3816:
3814:
3813:
3808:
3797:
3782:
3780:
3779:
3774:
3772:
3771:
3759:
3758:
3742:
3740:
3739:
3734:
3713:
3711:
3710:
3705:
3697:
3696:
3684:
3683:
3662:
3661:
3649:
3648:
3613:
3611:
3610:
3605:
3590:
3588:
3587:
3582:
3574:
3573:
3561:
3560:
3539:
3538:
3526:
3525:
3494:
3492:
3491:
3486:
3478:
3477:
3465:
3464:
3443:
3442:
3430:
3429:
3398:
3396:
3395:
3390:
3388:
3387:
3371:
3369:
3368:
3363:
3361:
3360:
3344:
3342:
3341:
3336:
3328:
3327:
3311:
3309:
3308:
3303:
3279:
3277:
3276:
3271:
3244:
3242:
3241:
3236:
3234:
3232:
3218:
3215:
3210:
3195:
3190:
3179:
3176:
3149:
3148:
3138:
3114:
3113:
3088:
3086:
3085:
3080:
3065:, such that for
3064:
3062:
3061:
3056:
3054:
3053:
3029:
3028:
3005:
3003:
3002:
2997:
2995:
2994:
2982:
2981:
2965:
2963:
2962:
2957:
2942:
2940:
2939:
2934:
2919:
2917:
2916:
2911:
2884:
2882:
2881:
2876:
2854:
2852:
2851:
2846:
2844:
2842:
2828:
2825:
2820:
2805:
2800:
2789:
2786:
2759:
2758:
2748:
2724:
2723:
2698:
2696:
2695:
2690:
2648:
2646:
2645:
2640:
2598:
2596:
2595:
2590:
2544:
2542:
2541:
2536:
2506:
2504:
2503:
2498:
2468:
2466:
2465:
2460:
2430:
2428:
2427:
2422:
2420:
2419:
2395:
2394:
2382:
2381:
2365:
2363:
2362:
2357:
2355:
2354:
2330:
2329:
2317:
2316:
2296:
2294:
2293:
2288:
2273:
2271:
2270:
2265:
2263:
2262:
2238:
2237:
2225:
2224:
2208:
2206:
2205:
2200:
2182:
2180:
2179:
2174:
2172:
2171:
2147:
2146:
2134:
2133:
2084:
2082:
2081:
2076:
2010:
2008:
2007:
2002:
1990:
1988:
1987:
1982:
1970:
1968:
1967:
1962:
1947:
1945:
1944:
1939:
1914:
1912:
1911:
1906:
1889:
1887:
1886:
1881:
1855:
1853:
1852:
1847:
1835:
1833:
1832:
1827:
1815:
1813:
1812:
1807:
1799:
1769:
1767:
1766:
1761:
1745:
1743:
1742:
1737:
1712:
1710:
1709:
1704:
1683:
1678:
1644:
1642:
1641:
1636:
1624:
1622:
1621:
1618:{\displaystyle }
1616:
1592:
1590:
1589:
1584:
1571:
1569:
1568:
1563:
1551:
1549:
1548:
1545:{\displaystyle }
1543:
1515:unified approach
1503:
1501:
1500:
1495:
1478:
1476:
1475:
1470:
1468:
1467:
1418:
1416:
1415:
1410:
1398:
1396:
1395:
1390:
1354:
1352:
1351:
1346:
1335:
1285:
1283:
1282:
1277:
1232:
1230:
1229:
1224:
1179:
1177:
1176:
1171:
1158:
1157:
1099:
1097:
1096:
1091:
1068:
1066:
1065:
1060:
1023:
1015:
1012:
1007:
943:as the limit of
942:
940:
939:
934:
891:
889:
888:
883:
881:
874:
872:
858:
855:
850:
835:
830:
819:
811:
807:
799:
794:
790:
789:
787:
776:
765:
762:
757:
731:
727:
719:
714:
710:
709:
705:
704:
701:
696:
693:
691:
687:
683:
681: applicants
680:
669:
666:
660:
658: applicants
657:
646:
643:
625:
620:
598:
587:
561:
557:
553:
552:
549:
544:
541:
528:
524:
523:
520:
515:
512:
510:
505:
502:
497:
494:
483:
478:
457:
453:
449:
448:
445:
440:
437:
432:
429:
424:
421:
410:
405:
273:
271:
270:
265:
253:
251:
250:
245:
240:
225:
223:
222:
217:
212:
196:
194:
193:
188:
183:
155:
153:
152:
147:
142:
111:
109:
108:
103:
67:marriage problem
51:optimal stopping
8473:
8472:
8468:
8467:
8466:
8464:
8463:
8462:
8433:Decision theory
8423:
8422:
8421:
8417:
8416:
8413:
8410:
8407:
8404:
8401:
8398:
8395:
8392:
8389:
8386:
8383:
8380:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8344:
8341:
8338:
8335:
8332:
8329:
8326:
8323:
8320:
8317:
8314:
8311:
8308:
8305:
8302:
8299:
8296:
8293:
8290:
8287:
8284:
8281:
8278:
8275:
8272:
8269:
8266:
8263:
8260:
8257:
8254:
8251:
8248:
8245:
8242:
8239:
8236:
8233:
8230:
8227:
8224:
8221:
8218:
8215:
8212:
8209:
8206:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8179:
8176:
8173:
8170:
8167:
8164:
8161:
8158:
8155:
8152:
8149:
8146:
8143:
8140:
8137:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8113:
8110:
8107:
8104:
8101:
8098:
8095:
8092:
8089:
8086:
8083:
8080:
8077:
8073:
8069:
8014:
8003:10.1.1.366.1718
7993:
7906:10.2307/2589677
7885:
7825:
7799:Pour la Science
7711:10.2307/2283044
7682:
7628:10.2307/1402748
7598:
7597:
7463:10.1.1.497.6468
7423:
7418:
7411:
7389:
7385:
7377:
7373:
7365:
7361:
7353:
7349:
7341:
7337:
7300:Cerebral Cortex
7292:
7288:
7265:10.1038/nrn2374
7249:
7245:
7200:
7196:
7141:
7137:
7129:
7125:
7117:
7110:
7102:
7095:
7087:
7083:
7067:
7066:
7031:
7027:
6996:
6992:
6953:
6949:
6912:
6908:
6893:
6889:
6854:
6850:
6839:
6835:
6827:
6823:
6815:
6811:
6802:
6800:
6798:
6772:
6768:
6760:
6756:
6748:
6744:
6734:
6732:
6722:
6718:
6712:Pour la Science
6665:
6661:
6632:
6621:
6617:
6574:
6567:
6560:
6553:
6550:
6529:
6526:
6525:
6502:
6499:
6498:
6495:bipartite graph
6477:
6474:
6473:
6470:
6462:Johannes Kepler
6451:F. Thomas Bruss
6420:Leonard Gillman
6408:
6392:anterior insula
6360:
6335:
6307:
6303:
6299:
6290:
6286:
6284:
6281:
6280:
6251:
6247:
6243:
6228:
6224:
6220:
6205:
6201:
6197:
6185:
6181:
6179:
6176:
6175:
6153:
6150:
6149:
6148:computed up to
6124:
6120:
6118:
6106:
6093:
6089:
6087:
6084:
6083:
6066:
6062:
6047:
6043:
6034:
6030:
6028:
6025:
6024:
6008:
6005:
6004:
5963:
5959:
5955:
5943:
5939:
5937:
5934:
5933:
5911:
5908:
5907:
5904:
5882:
5878:
5876:
5873:
5872:
5853:
5849:
5845:
5841:
5829:
5825:
5823:
5820:
5819:
5797:
5794:
5793:
5773:
5769:
5767:
5764:
5763:
5747:
5744:
5743:
5727:
5724:
5723:
5707:
5704:
5703:
5687:
5684:
5683:
5667:
5664:
5663:
5647:
5644:
5643:
5623:
5619:
5604:
5600:
5591:
5587:
5585:
5582:
5581:
5561:
5557:
5536:
5532:
5523:
5519:
5514:
5511:
5510:
5494:
5491:
5490:
5487:
5454:
5450:
5445:
5443:
5440:
5439:
5420:
5409:
5406:
5405:
5380:
5375:
5373:
5370:
5369:
5347:
5344:
5343:
5327:
5324:
5323:
5307:
5304:
5303:
5275:
5271:
5267:
5263:
5258:
5254:
5246:
5243:
5242:
5236:
5198:
5191:
5187:
5183:
5181:
5179:
5176:
5175:
5168:
5160:
5122:
5114:
5111:
5110:
5094:
5091:
5090:
5074:
5071:
5070:
5054:
5051:
5050:
5034:
5031:
5030:
5014:
5011:
5010:
4993:
4990:
4989:
4968:
4963:
4960:
4959:
4938:
4933:
4930:
4929:
4913:
4910:
4909:
4887:
4879:
4876:
4875:
4859:
4856:
4855:
4854:, we find that
4839:
4836:
4835:
4811:
4807:
4799:
4789:
4785:
4783:
4780:
4779:
4740:
4735:
4734:
4730:
4716:
4711:
4710:
4706:
4704:
4691:
4683:
4681:
4679:
4676:
4675:
4642:
4638:
4636:
4633:
4632:
4602:
4583:
4578:
4577:
4564:
4562:
4549:
4524:
4522:
4518:
4506:
4495:
4490:
4486:
4466:
4461:
4457:
4432:
4430:
4426:
4414:
4403:
4398:
4394:
4382:
4371:
4349:
4345:
4343:
4340:
4339:
4308:
4305:
4304:
4288:
4285:
4284:
4266:
4264:
4261:
4260:
4239:
4234:
4231:
4230:
4209:
4204:
4201:
4200:
4180:
4177:
4176:
4146:
4141:
4121:
4117:
4112:
4106:
4102:
4101:
4097:
4085:
4081:
4079:
4076:
4075:
4050:
4046:
4031:
4027:
4018:
4014:
4013:
4009:
3997:
3993:
3991:
3988:
3987:
3932:
3929:
3928:
3921:
3852:
3825:
3822:
3821:
3793:
3788:
3785:
3784:
3767:
3763:
3754:
3750:
3748:
3745:
3744:
3722:
3719:
3718:
3692:
3688:
3679:
3675:
3657:
3653:
3644:
3640:
3623:
3620:
3619:
3599:
3596:
3595:
3569:
3565:
3556:
3552:
3534:
3530:
3521:
3517:
3500:
3497:
3496:
3473:
3469:
3460:
3456:
3438:
3434:
3425:
3421:
3404:
3401:
3400:
3383:
3379:
3377:
3374:
3373:
3356:
3352:
3350:
3347:
3346:
3323:
3319:
3317:
3314:
3313:
3297:
3294:
3293:
3259:
3256:
3255:
3252:
3222:
3217:
3211:
3200:
3180:
3178:
3160:
3144:
3140:
3122:
3109:
3105:
3094:
3091:
3090:
3074:
3071:
3070:
3049:
3045:
3024:
3020:
3018:
3015:
3014:
2990:
2986:
2977:
2973:
2971:
2968:
2967:
2948:
2945:
2944:
2925:
2922:
2921:
2890:
2887:
2886:
2864:
2861:
2860:
2832:
2827:
2821:
2810:
2790:
2788:
2770:
2754:
2750:
2732:
2719:
2715:
2704:
2701:
2700:
2654:
2651:
2650:
2604:
2601:
2600:
2554:
2551:
2550:
2512:
2509:
2508:
2474:
2471:
2470:
2436:
2433:
2432:
2415:
2411:
2390:
2386:
2377:
2373:
2371:
2368:
2367:
2350:
2346:
2325:
2321:
2312:
2308:
2306:
2303:
2302:
2282:
2279:
2278:
2258:
2254:
2233:
2229:
2220:
2216:
2214:
2211:
2210:
2194:
2191:
2190:
2167:
2163:
2142:
2138:
2129:
2125:
2123:
2120:
2119:
2113:
2070:
2067:
2066:
2021:
1996:
1993:
1992:
1976:
1973:
1972:
1956:
1953:
1952:
1933:
1930:
1929:
1926:F. Thomas Bruss
1900:
1897:
1896:
1875:
1872:
1871:
1841:
1838:
1837:
1821:
1818:
1817:
1795:
1775:
1772:
1771:
1755:
1752:
1751:
1718:
1715:
1714:
1679:
1674:
1653:
1650:
1649:
1630:
1627:
1626:
1598:
1595:
1594:
1578:
1575:
1574:
1557:
1554:
1553:
1525:
1522:
1521:
1510:
1489:
1486:
1485:
1445:
1441:
1424:
1421:
1420:
1404:
1401:
1400:
1384:
1381:
1380:
1373:
1361:
1331:
1326:
1323:
1322:
1271:
1268:
1267:
1218:
1215:
1214:
1165:
1162:
1161:
1085:
1082:
1081:
1014:
1008:
1003:
979:
976:
975:
928:
925:
924:
879:
878:
862:
857:
851:
840:
820:
818:
809:
808:
798:
777:
766:
764:
758:
747:
742:
738:
729:
728:
718:
700:
694:applicant
692:
685:
684:
679:
665:
662:
661:
656:
642:
638:
635:
634:
630:
621:
610:
588:
577:
572:
568:
559:
558:
548:
542:applicant
540:
539:
535:
519:
513:applicant
511:
506:
501:
495:applicant
493:
492:
488:
479:
468:
455:
454:
444:
438:applicant
436:
428:
422:applicant
420:
419:
415:
406:
395:
384:
368:
366:
363:
362:
332:
280:
259:
256:
255:
236:
231:
228:
227:
208:
203:
200:
199:
179:
171:
168:
167:
138:
133:
130:
129:
97:
94:
93:
63:decision theory
17:
12:
11:
5:
8471:
8461:
8460:
8455:
8450:
8445:
8440:
8435:
8420:
8419:
8079:
8070:
8068:
8065:
8064:
8063:
8055:
8045:Neil Bearden.
8042:
8023:
8013:
8012:External links
8010:
8009:
8008:
7983:
7973:(4): 481–486.
7960:
7939:
7929:(3): 221–236.
7918:
7889:
7883:
7870:
7842:(2): 700–714.
7829:
7823:
7802:
7783:
7750:
7723:
7705:(313): 35–73.
7694:
7680:
7649:
7640:
7622:(2): 189–206.
7611:
7592:
7580:(3): 882–889.
7565:
7535:
7505:
7484:
7456:(5): 410–425.
7445:
7422:
7419:
7417:
7416:
7409:
7383:
7371:
7359:
7347:
7335:
7306:(4): 972–982.
7286:
7259:(6): 467–479.
7243:
7194:
7155:(2): 628–633.
7135:
7123:
7108:
7093:
7081:
7025:
6990:
6963:(2): 207–219.
6947:
6926:(3): 446–466.
6906:
6887:
6848:
6833:
6821:
6809:
6796:
6766:
6754:
6742:
6716:
6680:(2): 126–133.
6659:
6646:(3): 282–289.
6618:
6616:
6613:
6612:
6611:
6606:
6601:
6596:
6591:
6589:Odds algorithm
6586:
6580:
6579:
6565:
6549:
6546:
6533:
6506:
6481:
6469:
6466:
6435:Martin Gardner
6407:
6404:
6372:functional MRI
6359:
6356:
6334:
6331:
6314:
6311:
6306:
6302:
6298:
6293:
6289:
6258:
6255:
6250:
6246:
6242:
6235:
6232:
6227:
6223:
6219:
6212:
6209:
6204:
6200:
6196:
6191:
6188:
6184:
6163:
6160:
6157:
6132:
6127:
6123:
6115:
6112:
6109:
6105:
6101:
6096:
6092:
6069:
6065:
6061:
6058:
6055:
6050:
6046:
6042:
6037:
6033:
6012:
5992:
5989:
5986:
5983:
5980:
5977:
5970:
5967:
5962:
5958:
5954:
5949:
5946:
5942:
5921:
5918:
5915:
5903:
5900:
5885:
5881:
5856:
5852:
5848:
5844:
5840:
5837:
5832:
5828:
5807:
5804:
5801:
5776:
5772:
5751:
5731:
5711:
5691:
5671:
5651:
5626:
5622:
5618:
5615:
5612:
5607:
5603:
5599:
5594:
5590:
5569:
5564:
5560:
5556:
5553:
5550:
5547:
5544:
5539:
5535:
5531:
5526:
5522:
5518:
5498:
5486:
5483:
5467:
5464:
5461:
5457:
5453:
5449:
5427:
5423:
5419:
5416:
5413:
5402:Vanderbei 1980
5386:
5383:
5379:
5357:
5354:
5351:
5331:
5311:
5290:
5282:
5278:
5274:
5270:
5266:
5262:
5257:
5253:
5250:
5235:
5232:
5216:
5213:
5210:
5207:
5204:
5201:
5194:
5190:
5186:
5167:
5164:
5159:
5156:
5134:
5131:
5126:
5121:
5118:
5098:
5078:
5058:
5038:
5018:
4997:
4977:
4972:
4967:
4947:
4942:
4937:
4917:
4891:
4886:
4883:
4863:
4843:
4823:
4820:
4815:
4810:
4806:
4802:
4798:
4793:
4788:
4767:
4766:
4755:
4749:
4744:
4738:
4733:
4728:
4725:
4720:
4714:
4709:
4703:
4697:
4694:
4689:
4686:
4656:
4653:
4650:
4645:
4641:
4629:
4628:
4617:
4611:
4608:
4605:
4600:
4597:
4594:
4591:
4586:
4581:
4576:
4573:
4570:
4567:
4561:
4556:
4553:
4547:
4542:
4537:
4533:
4530:
4527:
4521:
4515:
4512:
4509:
4504:
4501:
4498:
4494:
4489:
4485:
4481:
4475:
4472:
4469:
4465:
4460:
4455:
4450:
4445:
4441:
4438:
4435:
4429:
4423:
4420:
4417:
4412:
4409:
4406:
4402:
4397:
4391:
4388:
4385:
4380:
4377:
4374:
4370:
4366:
4363:
4360:
4357:
4352:
4348:
4324:
4321:
4318:
4315:
4312:
4292:
4270:
4248:
4243:
4238:
4218:
4213:
4208:
4184:
4173:
4172:
4161:
4155:
4152:
4149:
4145:
4140:
4136:
4132:
4129:
4124:
4120:
4115:
4109:
4105:
4100:
4096:
4093:
4088:
4084:
4059:
4053:
4049:
4045:
4042:
4039:
4034:
4030:
4026:
4021:
4017:
4012:
4008:
4005:
4000:
3996:
3980:expected value
3936:
3920:
3917:
3901:
3900:
3893:
3886:
3851:
3848:
3835:
3832:
3829:
3806:
3803:
3800:
3796:
3792:
3770:
3766:
3762:
3757:
3753:
3732:
3729:
3726:
3703:
3700:
3695:
3691:
3687:
3682:
3678:
3674:
3671:
3668:
3665:
3660:
3656:
3652:
3647:
3643:
3639:
3636:
3633:
3630:
3627:
3603:
3580:
3577:
3572:
3568:
3564:
3559:
3555:
3551:
3548:
3545:
3542:
3537:
3533:
3529:
3524:
3520:
3516:
3513:
3510:
3507:
3504:
3484:
3481:
3476:
3472:
3468:
3463:
3459:
3455:
3452:
3449:
3446:
3441:
3437:
3433:
3428:
3424:
3420:
3417:
3414:
3411:
3408:
3386:
3382:
3359:
3355:
3334:
3331:
3326:
3322:
3301:
3269:
3266:
3263:
3251:
3248:
3231:
3228:
3225:
3221:
3214:
3209:
3206:
3203:
3199:
3193:
3189:
3186:
3183:
3175:
3172:
3169:
3166:
3163:
3159:
3155:
3152:
3147:
3143:
3137:
3134:
3131:
3128:
3125:
3121:
3117:
3112:
3108:
3104:
3101:
3098:
3078:
3069:stopping rule
3052:
3048:
3044:
3041:
3038:
3035:
3032:
3027:
3023:
2993:
2989:
2985:
2980:
2976:
2955:
2952:
2932:
2929:
2909:
2906:
2903:
2900:
2897:
2894:
2874:
2871:
2868:
2841:
2838:
2835:
2831:
2824:
2819:
2816:
2813:
2809:
2803:
2799:
2796:
2793:
2785:
2782:
2779:
2776:
2773:
2769:
2765:
2762:
2757:
2753:
2747:
2744:
2741:
2738:
2735:
2731:
2727:
2722:
2718:
2714:
2711:
2708:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2469:is changed to
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2418:
2414:
2410:
2407:
2404:
2401:
2398:
2393:
2389:
2385:
2380:
2376:
2353:
2349:
2345:
2342:
2339:
2336:
2333:
2328:
2324:
2320:
2315:
2311:
2286:
2261:
2257:
2253:
2250:
2247:
2244:
2241:
2236:
2232:
2228:
2223:
2219:
2198:
2170:
2166:
2162:
2159:
2156:
2153:
2150:
2145:
2141:
2137:
2132:
2128:
2112:
2109:
2108:
2107:
2104:
2093:
2092:
2089:
2086:
2074:
2029:Martin Gardner
2020:
2017:
2000:
1980:
1960:
1937:
1922:
1921:
1917:
1916:
1904:
1892:
1891:
1879:
1845:
1825:
1805:
1802:
1798:
1794:
1791:
1788:
1785:
1782:
1779:
1759:
1748:
1747:
1735:
1732:
1729:
1726:
1723:
1702:
1699:
1696:
1693:
1690:
1687:
1682:
1677:
1673:
1669:
1666:
1663:
1660:
1657:
1634:
1614:
1611:
1608:
1605:
1602:
1582:
1561:
1541:
1538:
1535:
1532:
1529:
1509:
1506:
1493:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1444:
1440:
1437:
1434:
1431:
1428:
1408:
1388:
1372:
1369:
1365:odds algorithm
1360:
1357:
1344:
1341:
1338:
1334:
1330:
1317:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1275:
1264:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1222:
1211:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1169:
1135:, the optimal
1108:is equal to 1/
1089:
1070:
1069:
1058:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1021:
1018:
1011:
1006:
1002:
998:
995:
992:
989:
986:
983:
932:
893:
892:
877:
871:
868:
865:
861:
854:
849:
846:
843:
839:
833:
829:
826:
823:
817:
814:
812:
810:
805:
802:
797:
793:
786:
783:
780:
775:
772:
769:
761:
756:
753:
750:
746:
741:
737:
734:
732:
730:
725:
722:
717:
713:
708:
699:
690:
678:
675:
672:
664:
663:
655:
652:
649:
641:
640:
637:
633:
629:
624:
619:
616:
613:
609:
605:
602:
597:
594:
591:
586:
583:
580:
576:
571:
567:
564:
562:
560:
556:
547:
538:
534:
531:
527:
518:
509:
500:
491:
487:
482:
477:
474:
471:
467:
463:
460:
458:
456:
452:
443:
435:
427:
418:
414:
409:
404:
401:
398:
394:
390:
387:
385:
383:
380:
377:
374:
371:
370:
331:
328:
315:
314:
310:
307:
304:
301:
298:
287:
279:
276:
263:
243:
239:
235:
215:
211:
207:
186:
182:
178:
175:
145:
141:
137:
126:odds algorithm
101:
15:
9:
6:
4:
3:
2:
8470:
8459:
8456:
8454:
8451:
8449:
8446:
8444:
8441:
8439:
8436:
8434:
8431:
8430:
8428:
8075:
8071:
8061:
8060:
8056:
8052:
8048:
8043:
8038:
8037:
8032:
8029:
8024:
8022:
8016:
8015:
8004:
7999:
7992:
7988:
7984:
7980:
7976:
7972:
7968:
7967:
7961:
7957:
7953:
7949:
7945:
7940:
7936:
7932:
7928:
7924:
7919:
7915:
7911:
7907:
7903:
7899:
7895:
7890:
7886:
7880:
7876:
7871:
7867:
7863:
7859:
7855:
7850:
7845:
7841:
7837:
7836:
7830:
7826:
7820:
7816:
7812:
7808:
7803:
7800:
7796:
7792:
7788:
7784:
7780:
7776:
7772:
7768:
7764:
7760:
7756:
7751:
7746:
7741:
7737:
7733:
7729:
7724:
7720:
7716:
7712:
7708:
7704:
7700:
7695:
7691:
7687:
7683:
7677:
7673:
7669:
7664:
7663:10.1.1.161.41
7659:
7655:
7650:
7646:
7641:
7637:
7633:
7629:
7625:
7621:
7617:
7612:
7608:
7602:
7593:
7588:
7583:
7579:
7575:
7571:
7566:
7561:
7556:
7552:
7548:
7544:
7540:
7536:
7531:
7526:
7522:
7518:
7514:
7511:(June 2000).
7510:
7506:
7502:
7498:
7494:
7490:
7485:
7481:
7477:
7473:
7469:
7464:
7459:
7455:
7451:
7446:
7442:
7438:
7434:
7430:
7425:
7424:
7412:
7406:
7402:
7398:
7394:
7387:
7380:
7379:Ferguson 1989
7375:
7368:
7363:
7356:
7351:
7344:
7339:
7331:
7327:
7322:
7317:
7313:
7309:
7305:
7301:
7297:
7290:
7282:
7278:
7274:
7270:
7266:
7262:
7258:
7254:
7247:
7239:
7235:
7230:
7225:
7221:
7217:
7213:
7209:
7205:
7198:
7190:
7186:
7181:
7176:
7171:
7166:
7162:
7158:
7154:
7150:
7146:
7139:
7133:
7127:
7120:
7115:
7113:
7105:
7100:
7098:
7090:
7085:
7077:
7071:
7063:
7059:
7055:
7051:
7047:
7043:
7039:
7035:
7029:
7021:
7017:
7013:
7009:
7006:(19): 51–65.
7005:
7001:
6994:
6986:
6982:
6978:
6974:
6970:
6966:
6962:
6958:
6951:
6943:
6939:
6934:
6929:
6925:
6921:
6917:
6910:
6902:
6898:
6891:
6883:
6879:
6875:
6871:
6867:
6863:
6859:
6852:
6844:
6837:
6830:
6825:
6818:
6813:
6799:
6793:
6789:
6785:
6781:
6777:
6770:
6763:
6758:
6751:
6746:
6731:
6727:
6720:
6713:
6709:
6703:
6699:
6695:
6691:
6687:
6683:
6679:
6675:
6674:
6669:
6663:
6654:
6649:
6645:
6641:
6637:
6630:
6628:
6626:
6624:
6619:
6610:
6607:
6605:
6604:Search theory
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6581:
6577:
6571:
6566:
6563:
6557:
6552:
6545:
6531:
6522:
6520:
6504:
6496:
6479:
6465:
6463:
6459:
6458:Arthur Cayley
6454:
6452:
6448:
6443:
6441:
6436:
6431:
6429:
6425:
6424:Samuel Karlin
6421:
6417:
6413:
6403:
6401:
6397:
6393:
6389:
6385:
6381:
6377:
6373:
6368:
6365:
6355:
6353:
6348:
6344:
6340:
6339:psychologists
6337:Experimental
6330:
6312:
6309:
6304:
6300:
6296:
6291:
6287:
6278:
6274:
6256:
6253:
6248:
6244:
6240:
6233:
6230:
6225:
6221:
6217:
6210:
6207:
6202:
6198:
6194:
6189:
6186:
6182:
6161:
6158:
6155:
6146:
6130:
6125:
6121:
6107:
6099:
6094:
6090:
6067:
6063:
6059:
6056:
6053:
6048:
6044:
6040:
6035:
6031:
6010:
5981:
5975:
5968:
5965:
5960:
5956:
5952:
5947:
5944:
5940:
5919:
5916:
5913:
5899:
5883:
5879:
5854:
5850:
5846:
5842:
5838:
5835:
5830:
5826:
5799:
5790:
5774:
5770:
5749:
5729:
5709:
5689:
5669:
5649:
5640:
5624:
5620:
5616:
5613:
5610:
5605:
5601:
5597:
5592:
5588:
5562:
5558:
5554:
5551:
5548:
5545:
5542:
5537:
5533:
5529:
5524:
5520:
5496:
5482:
5465:
5462:
5459:
5455:
5451:
5447:
5425:
5421:
5417:
5414:
5411:
5403:
5384:
5381:
5377:
5349:
5329:
5309:
5288:
5280:
5276:
5272:
5268:
5264:
5260:
5255:
5251:
5248:
5239:
5231:
5211:
5208:
5205:
5199:
5192:
5188:
5184:
5173:
5163:
5155:
5152:
5146:
5132:
5129:
5124:
5119:
5116:
5096:
5076:
5056:
5036:
5016:
4995:
4970:
4940:
4915:
4908:is convex in
4907:
4889:
4884:
4881:
4861:
4841:
4821:
4818:
4813:
4808:
4800:
4796:
4791:
4771:
4753:
4747:
4742:
4736:
4731:
4726:
4723:
4718:
4712:
4707:
4701:
4695:
4687:
4674:
4673:
4672:
4670:
4651:
4643:
4639:
4615:
4609:
4606:
4603:
4598:
4595:
4592:
4589:
4584:
4579:
4574:
4571:
4568:
4565:
4559:
4554:
4551:
4545:
4540:
4535:
4531:
4528:
4525:
4519:
4513:
4510:
4507:
4502:
4499:
4496:
4492:
4487:
4483:
4479:
4473:
4470:
4467:
4463:
4458:
4453:
4448:
4443:
4439:
4436:
4433:
4427:
4421:
4418:
4415:
4410:
4407:
4404:
4400:
4395:
4389:
4386:
4383:
4378:
4375:
4372:
4368:
4364:
4358:
4350:
4346:
4338:
4337:
4336:
4322:
4319:
4316:
4313:
4310:
4290:
4268:
4241:
4211:
4198:
4182:
4159:
4153:
4150:
4147:
4143:
4138:
4134:
4130:
4127:
4122:
4118:
4107:
4103:
4098:
4094:
4091:
4086:
4082:
4074:
4073:
4072:
4057:
4051:
4047:
4043:
4040:
4037:
4032:
4028:
4024:
4019:
4015:
4010:
4003:
3998:
3994:
3985:
3981:
3976:
3973:
3969:
3965:
3961:
3957:
3953:
3950:
3934:
3925:
3916:
3914:
3910:
3906:
3898:
3894:
3891:
3887:
3884:
3881: =
3880:
3875:
3871:
3870:
3869:
3867:
3859:
3855:
3847:
3833:
3830:
3827:
3818:
3804:
3801:
3798:
3794:
3790:
3768:
3764:
3760:
3755:
3751:
3730:
3727:
3724:
3715:
3693:
3689:
3685:
3680:
3676:
3666:
3658:
3654:
3650:
3645:
3641:
3628:
3625:
3617:
3601:
3592:
3570:
3566:
3562:
3557:
3553:
3543:
3535:
3531:
3527:
3522:
3518:
3505:
3502:
3474:
3470:
3466:
3461:
3457:
3447:
3439:
3435:
3431:
3426:
3422:
3409:
3406:
3384:
3380:
3357:
3353:
3332:
3329:
3324:
3320:
3299:
3291:
3287:
3283:
3267:
3264:
3261:
3246:
3229:
3226:
3223:
3219:
3212:
3207:
3204:
3201:
3197:
3191:
3187:
3184:
3181:
3173:
3170:
3167:
3164:
3161:
3153:
3145:
3141:
3135:
3132:
3129:
3126:
3123:
3115:
3110:
3106:
3099:
3096:
3076:
3068:
3050:
3046:
3042:
3039:
3036:
3033:
3030:
3025:
3021:
3010:
3007:
2991:
2987:
2983:
2978:
2974:
2953:
2950:
2930:
2927:
2904:
2901:
2898:
2892:
2872:
2869:
2866:
2856:
2839:
2836:
2833:
2829:
2822:
2817:
2814:
2811:
2807:
2801:
2797:
2794:
2791:
2783:
2780:
2777:
2774:
2771:
2763:
2755:
2751:
2745:
2742:
2739:
2736:
2733:
2725:
2720:
2716:
2709:
2706:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2548:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2416:
2412:
2408:
2405:
2402:
2399:
2396:
2391:
2387:
2383:
2378:
2374:
2351:
2347:
2343:
2340:
2337:
2334:
2331:
2326:
2322:
2318:
2313:
2309:
2300:
2297:for Bob is a
2284:
2275:
2259:
2255:
2251:
2248:
2245:
2242:
2239:
2234:
2230:
2226:
2221:
2217:
2196:
2189:
2188:stopping rule
2184:
2168:
2164:
2160:
2157:
2154:
2151:
2148:
2143:
2139:
2135:
2130:
2126:
2118:
2105:
2103:to trick Bob.
2102:
2098:
2097:
2096:
2090:
2087:
2072:
2064:
2063:
2062:
2060:
2059:zero-sum game
2056:
2050:
2048:
2042:
2041:
2039:
2034:
2030:
2026:
2016:
2012:
1998:
1978:
1958:
1949:
1935:
1927:
1919:
1918:
1902:
1894:
1893:
1877:
1869:
1868:
1867:
1866:
1861:
1859:
1843:
1823:
1803:
1800:
1796:
1792:
1789:
1783:
1777:
1770:be such that
1757:
1733:
1730:
1727:
1724:
1721:
1700:
1697:
1691:
1685:
1680:
1675:
1671:
1667:
1661:
1655:
1648:
1647:
1646:
1632:
1609:
1606:
1603:
1580:
1559:
1536:
1533:
1530:
1518:
1516:
1505:
1491:
1482:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1438:
1435:
1432:
1426:
1406:
1386:
1377:
1368:
1366:
1356:
1342:
1339:
1336:
1332:
1328:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1273:
1266:
1265:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1220:
1213:
1212:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1167:
1160:
1159:
1156:
1154:
1150:
1146:
1142:
1138:
1134:
1129:
1127:
1123:
1119:
1115:
1111:
1107:
1103:
1087:
1079:
1075:
1056:
1049:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1019:
1016:
1009:
1004:
1000:
996:
993:
987:
981:
974:
973:
972:
970:
966:
962:
958:
954:
950:
946:
930:
922:
918:
914:
910:
906:
902:
898:
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359:
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349:
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341:
337:
336:stopping rule
327:
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320:
311:
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299:
296:
292:
288:
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284:
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127:
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116:
115:stopping rule
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88:
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80:
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72:
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64:
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8074:
8058:
8051:the original
8034:
7970:
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7947:
7943:
7926:
7922:
7897:
7893:
7874:
7839:
7833:
7806:
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7790:
7785:Hill, T.P. "
7762:
7758:
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7432:
7428:
7392:
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7362:
7357:, Problem 3.
7355:Gardner 1966
7350:
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7303:
7299:
7289:
7256:
7252:
7246:
7211:
7207:
7197:
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7126:
7084:
7070:cite journal
7045:
7041:
7028:
7003:
6999:
6993:
6960:
6956:
6950:
6923:
6919:
6909:
6900:
6897:Scripta Math
6896:
6890:
6865:
6861:
6851:
6836:
6829:Bearden 2006
6824:
6812:
6801:, retrieved
6779:
6769:
6762:Gardner 1966
6757:
6745:
6733:. Retrieved
6729:
6719:
6711:
6677:
6671:
6662:
6643:
6639:
6523:
6471:
6455:
6446:
6444:
6432:
6427:
6409:
6369:
6364:neuroscience
6361:
6336:
6275:
6147:
5905:
5818:limit, each
5791:
5641:
5488:
5240:
5237:
5169:
5161:
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4905:
4777:
4668:
4630:
4196:
4174:
4071:is given by
3983:
3977:
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3967:
3963:
3951:
3926:
3922:
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3908:
3904:
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3896:
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3882:
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3873:
3864:
3853:
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3716:
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3593:
3372:, else pick
3345:, then pick
3253:
3066:
3012:
3008:
2857:
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2185:
2114:
2094:
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2052:
2046:
2044:
2036:
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1950:
1923:
1865:1/e-strategy
1864:
1862:
1858:1/e-strategy
1857:
1749:
1519:
1514:
1511:
1480:
1378:
1374:
1362:
1320:
1152:
1148:
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1125:
1121:
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1109:
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1101:
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920:
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322:
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157:
123:
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86:
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66:
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8405:print_table
8396:to_markdown
8369:'n'
8267:print_table
7950:: 140–152.
7795:cover story
7765:: 226–240.
7048:(1): 3–13.
6817:Gnedin 1994
6750:Gnedin 2021
6708:cover story
4671:, one gets
1371:Limitations
278:Formulation
79:googol game
8427:Categories
7996:(Report).
7900:(3): 215.
7421:References
7367:Bruss 1984
7343:Flood 1958
6903:: 289–292.
6735:6 February
6497:where the
6343:economists
4199:is either
313:otherwise.
289:There are
81:, and the
59:statistics
8387:transpose
8300:DataFrame
8036:MathWorld
7998:CiteSeerX
7849:1204.5537
7779:245449000
7658:CiteSeerX
7458:CiteSeerX
7062:2299-4009
7020:0137-2890
6985:121339045
6977:0022-3239
6942:0022-247X
6882:0025-1909
6730:Big Think
6702:124798270
6694:1545-2786
6521:problem.
6440:Leo Moser
6305:−
6249:−
6226:−
6203:−
6187:−
6114:∞
6111:→
6057:⋯
5988:∞
5985:→
5961:−
5945:−
5847:−
5836:∼
5806:∞
5803:→
5614:⋯
5356:∞
5353:→
5209:−
5130:−
4976:⌉
4966:⌈
4946:⌋
4936:⌊
4805:∂
4787:∂
4708:−
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4685:∂
4596:−
4575:−
4529:−
4511:−
4493:∏
4437:−
4419:−
4401:∏
4387:−
4369:∑
4320:≤
4314:≤
4247:⌉
4237:⌈
4217:⌋
4207:⌊
4041:…
3805:ϵ
3725:ϵ
3629:∈
3506:∈
3227:−
3198:∑
3185:−
3165:∈
3154:≤
3127:∈
3111:τ
3077:τ
2928:−
2837:−
2808:∑
2795:−
2775:∈
2737:∈
2721:τ
2285:τ
2197:τ
1999:τ
1979:τ
1844:τ
1824:τ
1784:τ
1758:τ
1731:≤
1725:≤
1672:∫
1625:and let
1465:⋯
1340:≈
1044:
1035:−
1001:∫
867:−
838:∑
825:−
796:⋅
782:−
771:−
745:∑
716:⋅
674:−
651:−
608:∑
593:−
575:∑
530:⋅
466:∑
434:∩
393:∑
319:candidate
297:is known.
174:∼
7989:(2012).
7866:31778896
7435:: 58–9.
7330:24142842
7273:18464792
7238:12417672
6548:See also
6380:parietal
6082:, where
5580:, where
5289:⌋
5256:⌊
5151:learning
4904:. Since
3820:But for
3410:∉
3288:and the
3286:T. Cover
3250:Solution
2025:Ferguson
1517:(1984):
951:, using
903:(1) = 1/
87:37% rule
8312:columns
7914:2589677
7801:(2009))
7719:2283044
7690:2742443
7636:1402748
7480:9186039
7321:4366612
7281:7416645
7229:6758024
7189:8570606
7157:Bibcode
6803:25 June
6714:(2009).
6406:History
6313:1474560
6310:4162637
6174:, with
5089:, then
3982:of the
3958:from a
3282:minimax
156:(where
8258:values
8249:return
8237:values
8231:argmax
8213:values
8156:return
8144:return
8096:pandas
8093:import
8081:import
8000:
7912:
7881:
7864:
7821:
7777:
7717:
7688:
7678:
7660:
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6983:
6975:
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6880:
6794:
6700:
6692:
6445:The 1/
6416:Purdue
6400:reward
6394:, and
4778:Since
3972:payoff
3956:i.i.d.
3954:drawn
2085:cards.
2047:googol
1315:0.399
1312:0.406
1309:0.410
1306:0.414
1303:0.428
1300:0.433
1297:0.458
1294:0.500
1291:0.500
1288:1.000
967:for 1/
77:, the
73:, the
69:, the
61:, and
8375:print
8357:index
8339:n_max
8327:range
8321:index
8273:n_max
8252:r_max
8219:r_max
8201:solve
8084:numpy
8067:Notes
7994:(PDF)
7910:JSTOR
7862:S2CID
7844:arXiv
7775:S2CID
7715:JSTOR
7686:S2CID
7632:JSTOR
7476:S2CID
7277:S2CID
7180:40102
6981:S2CID
6698:S2CID
6615:Notes
5906:When
5404:, if
5400:. By
3312:. If
2547:as if
1343:0.368
957:(i−1)
945:(r−1)
8363:name
8306:data
8282:data
8150:else
8111:func
8018:OEIS
7879:ISBN
7819:ISBN
7676:ISBN
7607:link
7405:ISBN
7326:PMID
7269:PMID
7234:PMID
7185:PMID
7076:link
7058:ISSN
7016:ISSN
6973:ISSN
6938:ISSN
6878:ISSN
6805:2023
6792:ISBN
6737:2024
6690:ISSN
6382:and
6374:. A
6341:and
6257:1152
6254:2761
5617:>
5611:>
5598:>
5185:0.25
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3968:each
3964:must
3831:>
3728:>
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3254:For
2055:game
1863:The
1750:Let
963:and
955:for
323:Skip
45:The
8399:())
8264:def
8198:def
8189:sum
8108:def
7975:doi
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7948:151
7931:doi
7902:doi
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7854:doi
7811:doi
7789:".
7767:doi
7763:145
7740:doi
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7668:doi
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7555:doi
7525:doi
7497:doi
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7437:doi
7397:doi
7316:PMC
7308:doi
7261:doi
7224:PMC
7216:doi
7175:PMC
7165:doi
7050:doi
7008:doi
6965:doi
6928:doi
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6784:doi
6682:doi
6648:doi
6104:lim
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5682:to
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4229:or
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