Game theory to a friend’s rescue

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Abstract

A friend asked us to determine good strategies for an interesting discrete search game. We were able to find one that helped him.

Introduction

A friend of ours usually plays the following game to see whether he or his favourite dinner companion will pick up the tab:

Player 1 (the “Hider”) makes the first move by selecting a target number (an integer) between 1 and 999 inclusive and then writing it on a piece of paper making sure that Player 2 (the “Searcher”) does not see it. Then the Searcher selects a series of numbers trying to get the Hider to call the target number. The Hider’s rules are quite straightforward. Whatever integer the Searcher calls, the Hider has to choose the integer adjacent to the Searcher’s number and closest to the target number. For instance suppose that the Hider’s target number is 589 and the Searcher begins by selecting 614. Then the Hider would be obligated to call 613 because it is the integer adjacent to 614 and closest to 589. At this point the Searcher knows the target number is in the range 1–612. Suppose that the Searcher selects 411 next. Then the Hider would call 412 and the game would continue in this way until one of the players selects the target number. The Searcher must always call a number within the reduced range determined from the Hider’s responses. Finally, the player who calls the target number has to pay for dinner.

We term this game the Hider–Searcher Game. Normally the players alternate roles in the game, so that, in the long-run, each of them will play the Hider and Searcher roles about the same number of times.

When our friend came to us asking for help, he was not doing very well. He had lost over 80% of the time and had accumulated over $900 in net losses. He wondered if we could come up with a strategy that would help him win more often. He had tracked the numbers his dinner companion had chosen as the Hider and wanted to know if there was a pattern in these numbers he could take advantage of.

Rather than look for a pattern in his opponent’s choices, we decided to take a game theoretic approach. Based on our informal observations of the way people play the game, most Hiders choose a target somewhere in the “middle” and then the Searcher sets out to make the Hider call this number. For the purposes of this paper, we define the Middle Strategy to be the case where a Hider chooses a target number from the set {3,4,,996,997}.

Intuitively, it seems reasonable that the Middle Strategy is the best way to go. It provides protection in that the target number is surrounded by lots of numbers and the Searcher would be quite lucky to choose one of the two numbers adjacent to the target with his first choice. Equivalently, choosing an end-pair number (the end pairs are {1,2} or {998,999}) seems to “expose” the Hider.

But these arguments are incorrect. As it turns out, the Middle Strategy is not the best strategy. It guarantees the Hider will lose about 2/3 of the time. The best Hider strategy is to first choose one of the end-pairs at random and then choose one of the two numbers in that end-pair at random. This assures the Hider a 50–50 chance of winning vice the 1/3 chance if the Middle Strategy were followed.

This Hider–Searcher Game is a discrete search game. The reader is referred to Gal  [2], [3], Garnaev  [4], and Haley and Stone  [6] for reviews of the search game literature. The search games that are most similar to ours are those offered by Gal  [1] and Gilbert  [5]. They study games where a player writes down an integer drawn from the first n integers and the other tries to guess the selected integer. For each guess, the player is provided information about where the “hidden” integer is in relation to the guess. The primary difference between ours and theirs is the payoff structure. Whereas their games offer a payoff proportional to the number of iterations to guess the integer, ours provides a single payment unrelated to the number of iterations.

Section snippets

What happens when the Hider plays the Middle Strategy

Let the choice set be Γ={1,2,,999} and suppose the Hider plays the Middle Strategy by choosing a number at random from the set ΓM={3,4,,996,997}. In response, we suppose that the Searcher chooses 1 or 2 at random and then every third integer after that. So the Searcher’s sequence is either S1={1,4,7,,997} or S2={2,5,8,,998}. We term this strategy 3Search(1, 2). The rationale for “every third integer” is as follows. Suppose the first two calls of a game instance are: Searcher :2Hider :3. At

The equilibrium strategy

We are interested in equilibria for the Hider–Searcher Game over the choice set Γ. Suppose the Hider chooses the target number, t0, from Γ using the mixed strategy p=(p1,p2,,p999) where pi is the probability that t0 takes the integer i. Similarly, let the Searcher’s first integer, s0, be chosen with the mixed strategy q=(q1,q2,,q999) where qi is the probability of picking integer i. Let the payoff matrix be P. We assume it to have elements pts where t is the pure strategy choice of the Hider

How non-game theorists play

We ran an experiment with an undergraduate OR class where game theory was taught. Students were introduced to the Hider–Searcher Game in the first period of the course. Their project was to develop good strategies for the game and use them in a tournament that all students would participate in at the end of term. For the tournament, the students were randomly paired and each played the role of Hider 5 times and Searcher 5 times. The game was played over the reduced choice set ΓE={1,2,,99}.

The

Some comments on fairness

In the case where both players play optimal strategies, the game reduces to flipping a coin for dinner. There is a case for arguing that the game is boring if players begin to play the optimal strategies. In other words, what makes the game interesting – the hide and seek over a large set of numbers – disappears. That said, our experimental evidence suggests that optimal play is not going to start any time soon.

Even if two people play suboptimally, the game is fair as long as the players

The game on a tree

The game we propose may be modelled as a search along a path where the nodes are consecutively numbered from 1 to 999: 1  —2—  3  —   —  998  —  999. At each iteration, the Searcher selects a number, and if it is not the target number, the Hider then calls out the adjacent node closest to the target number. This has the effect of pruning the path so that what remains is a smaller sub-path containing the target number. (The adjacent node and those either to the left or right of it are

Back to our friend

We explained the equilibrium strategies to our friend and he has begun to use them. His opponent uses the Middle Strategy. Hence our friend should win 1/2 the time when he is the Hider and 2/3 of the time when he is the Searcher. His weighted probability of winning is 12(0.50)+12(0.67)=0.583. Since our tutoring, his performance has improved. Of the last 11 plays of the game, he has won 8 times.

Conclusion

In this paper, we have described a game that some of our friends use to determine who pays for dinner. It is an interesting game because its optimal strategies are counter-intuitive. We see the main contribution of the paper to be an application of game theory, a theoretical area where there are not many direct applications.

Acknowledgement

We thank an anonymous referee for suggesting an extension of this game to a tree.

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There are more references available in the full text version of this article.

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