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Rationality and Game TheoryThe discrepancy between the predictions that game theory makes for the way to play the centipede game and what happens in actual practice has created a torrent of research...
If you were offered $1000, no strings attached, you would probably take the money. And run? If the amount were $10,000 rather than $1000, again, no strings attached, you would probably take the money, and run even faster. However, if you saw a $100 bill lying on the street, and your back had been giving you agonizing pain recently, though currently you were in no pain, you might not bend over to pick it up. Certainly for a quarter (25 cents) it would not be worth it to stoop over. In explaining and offering advice about economic behavior, economists and mathematicians invoke arguments about "rational" behavior to explain what actions people should take. However, as the example above shows, if the decision maker has more information (the state of her back) than the observer, then the behavior observed may be different from the behavior expected.
Game Theory: The Basics
Game theory has grown into a complex and multi-branched subject. The basic idea is that one has a group of people (usually referred to as players) who are interacting with each other with some conflict the group desires to resolve. For simplicity, let us consider only two people (countries or companies) as players. Depending on the actions or decisions that are taken by the players, different "payoffs" will accrue to the two people involved. We will assume that the games involved are games of perfect information. This means that each player knows exactly what actions are open to him/her and to his/her opponent. It is also assumed that the payoffs to each player are known and what the value of each payoff is. The games we are interested in here typically arise in daily life, economics and political science (law), in contrast to games such as Nim, Dots and Boxes, and Hackenbush, which are sometimes called combinatorial games. The examples below give the range of conflict situations that game theory illuminates.
This brief discussion should suggest that "utility theory" is a fascinating and complex subject. Thus, a rich and a poor person would look at the same sum of money that might change hands in a game quite differently. For our purposes let us pretend that the players are able to evaluate for themselves how to measure the outcomes of the game on some scale of utilities. These outcomes of the game arise from the players making a choice among actions that are available to them. An action on the part of each player leads to an outcome. Sometimes the utilities associated with an outcome will be known to both of the players and sometimes they won't. We will also assume that given a choice of a higher versus lower numerical payoff, both players will always select the higher payoff.
However, many of the most interesting games that mathematicians and those who try to apply game theory in the real world have to analyze are not zero-sum games. For example the game below might be a model of the situation described earlier in which a couple is trying to decide whether to go to a movie or to the opera.
For those with mathematical skill one can master the techniques for finding the optimal play of games such as the ones we have been looking at. It would be nice, however, if there was a more pragmatic approach to playing a game such as the one we solved (Figure 6). An insight into this was obtained by the mathematician Julia Robinson (1918-1985), who is much better known for her famous role in the solution of Hilbert's 10th Problem. Robinson was also the first woman President of the American Mathemtical Society. She proved an intriguing result based on an idea pioneered by the Princeton educated mathematician George William Brown. Brown introduced the idea of "fictitious play," though the terminology is not particularly felicitous because it is a technique that often can be used in practice to play games.
Perhaps the most famous of all games is Prisoner's Dilemma. It deserves (and perhaps will eventually get) a column of its own. The game takes its name from a "story" to accompany the matrix of the game, that is due to Albert W. Tucker, who for many years headed the Mathematics Department at Princeton University. The story helps explain the payoffs of a 2-person nonzero-sum game with a structure that has "paradoxical" implications.
(Albert W. Tucker, courtesy of his son Alan Tucker)
The Centipede Game
Compared with Prisoner's Dilemma the centipede game is not that well known, but like Prisoner's Dilemma, it has been a very fertile window into gaining insight into the nature of rationality. The discrepancy between the predictions that game theory makes for the way to play the centipede game and what happens in actual practice has created a torrent of research.
(Courtesy of Professor Randall Ellis, Dept. of Economics, Boston University)
Suppose you are A. How would you play this game? Here is the mathematical reasoning that comes with treating this game as one of perfect information, where both players are completely rational each opponent behaves rationally. Suppose the game were to reach the stage where player B has to act at the far right decision vertex. Now B, preferring more money rather than less will choose take (down) at this node. This follows since if B passes, he gets 16 while his opponent A gets 64. By "taking," B gets 32 while A only gets 8. Thus, at the prior vertex, A knowing what B will do, will chose to pick take (down). Knowing this, "backward reasoning" means that at the prior node B will take, which in turn means that at the very first decision node, A should take as well! Reasoning of this kind has come to be called "backward induction." Similarly, in the "long" (e.g. many feet) version of the centipede game it means that A should choose the take (down) option on the very first play of the game. This means that both players receive relatively meager payoffs. To play longer offers each of the players a chance to make more but only at the risk that when the game is stopped, his/her opponent will walk away with more than he/she will.
In this variant of the game the amounts each player gets are going up quickly; however, in some variants, the growth is very slow. Furthermore, in some variants the initial pot, if taken, gives each equal amounts. One can look at a version of the game where the sum of the payoffs to the players is a constant. The characteristic of these games is that rational play suggests that the first player should take right at the start. Yet, this seems a perplexing choice since so much more can be gained by being "irrational." Also, note that if A passes (allows B to move) at the first node of the game, then B faces a new version of the centipede game with essentially the same structure that A had! Why might you not take at the first opportunity if you are player A? One reason is that you might have some "altruistic" component to your personality. You might also think that perhaps your opponent, if the centipede is longer (rather than having 4 legs), will make a mistake in the analysis and continue to play longer when it is his/her decision.
Are you surprised that many people do not play this game "rationally" in the sense that when given the chance to play A at the first round, they do not choose "take"? On the other hand it is also the case that the game does not typically last as long as allowing B to make the pass move at the last vertex. Variants of the centipede game have provided a controlled environment to probe the nature of altruism, signaling to achieve cooperation, and the limits to rationality. These experiments have worked with the centipede game in extensive form as well as in recent work that gets new insights from using a normal form for the centipede game. In this work the subjects of the experiments must name the "round" where they would "take" in the centipede game rather than play the game dynamically as such. An interesting recent set of experiments concerns playing the centipede game where groups of players compete in the game against each other rather than individuals playing each other. The results here suggest that groups come closer to the prediction of "taking" on the first round of play than when individuals play. The accelerating number of papers of both a theoretical type and experimental type that draw inspiration from the centipede game are a testament to the fertility of a single game and its associated circle of ideas to stimulate mathematicians and scholars over a wide array of disciplines.
NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.
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