GAME THEORY

Introduction

branch of applied mathematics fashioned to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests. In a typical game, decision-making “players,” who each have their own goals, try to outsmart one another by anticipating each other's decisions; the game is finally resolved as a consequence of the players' decisions. A solution to a game prescribes the decisions the players should make and describes the game's appropriate outcome.

Game theory serves as a guide for players and as a tool for predicting the outcome of a game. Although game theory may be used to analyze ordinary parlour games, its range of application is much wider. In fact, game theory was originally designed by the Hungarian-born American mathematician John von Neumann and his colleague Oskar Morgenstern, a German-born American economist, to solve problems in economics. In their book The Theory of Games and Economic Behavior, published in 1944, von Neumann and Morgenstern asserted that the mathematics developed for the physical sciences, which describes the workings of a disinterested nature, was a poor model for economics. They observed that economics is much like a game in which the players anticipate one another's moves and that it therefore requires a new kind of mathematics, which they appropriately named game theory.

Game theory may be applied in situations in which decision makers must take into account the reasoning of other decision makers. By stressing strategic aspects—aspects controlled by the participants rather than by pure chance—the method both supplements and goes beyond the classical theory of probability. It has been used, for example, to determine the formation of political coalitions or business conglomerates, the optimum price at which to sell products or services, the power of a voter or a bloc of voters, the selection of a jury, the best site for a manufacturing plant, and even the behaviour of certain species in the struggle for survival.

It would be surprising if any one theory could address such a wide range of “games,” and, in fact, there is no single “game theory.” A number of theories exist, each applicable to a different kind of situation and each with its own kind of solution (or solutions).

This article discusses some of the simpler games and theories as well as the basic principles involved in game theory. Additional techniques and concepts that may be used in solving decision problems are treated in optimization. For information pertaining to the classical theory of probability, see the articles mathematics, history of; and probability theory.

 

Classification of games

Games are grouped into several categories according to certain significant features, the most obvious of which is the number of players involved. A game can thus be designated as being one-person, two-person, or n-person (with n larger than two), and the games in each category have their own distinct natures. A player need not be a single person, of course; it may be a nation, a corporation, or a team consisting of many people with identical interests relative to

the game. In games of perfect information, such as chess, each player knows everything about the game at all times. Poker, on the other hand, is an example of a game of imperfect information because players do not know the cards the other players are dealt.

The extent to which the goals of the players are opposed or coincide is another basis for classifying games. Zero-sum (or, more accurately, constant-sum) games are completely competitive. Poker, for example, is a zero-sum game because the combined wealth of the players remains constant; if one player wins, another must lose because money is neither created nor destroyed. Players in zero-sum games have completely conflicting interests. In nonzero-sum games, however, all the players can be winners (or losers). In a labour-management dispute, for example, the two parties have some conflicting interests yet both may benefit if a strike is avoided.

Nonzero-sum games can be further distinguished as being either cooperative or noncooperative. In cooperative games players may communicate and make binding agreements in advance; in noncooperative games they may not. An automobile salesman and a potential customer are engaged in a cooperative game, as is a business and its employees; participants bidding independently at an auction are playing a noncooperative game.

Finally, a game is said to be finite when each player has a finite number of decisions to make and has only a finite number of alternatives for each decision. Chess, checkers, poker, and most parlour games are finite. Infinite games, in which there are either an infinite number of alternatives or an infinite number of decisions, are much subtler and more complicated. They are discussed only briefly in this article.

A game can usually be described in one of three ways: in extensive, normal, or characteristic-function form. Most parlour games, which progress step by step, a move at a time, are described in extensive form; that is, they are represented by a “tree.” Each step or position is represented by a vertex of the tree, and the branches connecting the vertices represent the players' alternatives or moves. The normal form is used primarily to describe two-person games. In this form a game is represented by a matrix in which each column is associated with a strategy of one player and each row with a strategy of the second player; the matrix entry in a particular row and column is the outcome of the game if the two associated strategies are used. The normal form is important theoretically and can also be used in practice if the number of available strategies is small. The characteristic-function form, which is generally used only for games with more than two players, indicates the minimum value each coalition of players (including single-player coalitions) can obtain when playing against a coalition made up of all the remaining players.

 

One-person games

In the one-person game, the simplest game of all, there is only one decision maker. Because he has no opponents to thwart him, the player need only list the options available to him and then choose one. If chance is involved, the game might seem to be more complicated, but in principle the decision is still relatively simple. A man deciding whether to carry an umbrella, for example, weighs the risks involved and makes his choice. He may make the wrong decision, but he need not be worried about being outsmarted by other players; that is, he need

not take into account the decisions of others. One-person games, therefore, hold little interest for game theoreticians.

 

Two-person zero-sum games

Games of perfect information

The simplest game of any real theoretical interest is the finite two-person zero-sum game of perfect information. Examples of such games include chess, checkers, and the Japanese game go. In 1912 Ernst Zermelo proved that such games are strictly determined; this means that rational players making use of all available information can deduce a strategy that is clearly optimal and so the outcome of such games is preordained. In chess, for example, exactly one of three possibilities must be true: (1) white has a winning strategy (one that wins against any strategy of black); (2) black has an analogous winning strategy; or (3) white and black each have a strategy that guarantees them a win or a draw. (Proper play by both white and black leads to a draw.) Because a sufficiently rapid computer could analyze such games completely, they are of only minor theoretical interest.

 

Games of imperfect information

The simplest two-person zero-sum games of imperfect information are those that have saddle points. (All two-person zero-sum games of perfect information have saddle points.) Such games have a predetermined outcome (assuming rational play), and each player can, by choosing the right strategy, obtain an amount at least equal to this outcome no matter what the other player does. This predetermined outcome is called the value of the game. An example of such a game is described in normal form below.

Two campaigning political parties, A and B, must each decide how to handle a controversial issue in a certain town. They can either support the issue, oppose it, or evade it. Each party must make its decision without knowing what its rival will do. Every pair of decisions determines the percentage of the vote that each party receives in the town, and each party wants to maximize its own percentage of the vote. The entries in the matrix represent party A's percentage (the remaining percentage goes to party B); if, for example, A supports the issue and B evades it, A gets 80 percent (and B, 20 percent) of the vote.

A's decision seems difficult at first because it depends upon B's strategy. A does best to oppose if B supports, evade if B opposes, and support if B evades. A must therefore consider B's decision before making its own. No matter what A does, B gains the largest percentage of votes by opposing the issue. Once A recognizes this, its strategy should clearly be to evade and settle for 30 percent of the vote. This 30 percent/70 percent division of the vote is the game's saddle point.

A more systematic way of finding the saddle point is to determine the maximin and minimax values. Using this method, A first determines the minimum percentage of votes it can obtain for each of its strategies and then finds the maximum of these three minimum values. The minimum percentages A will get if it supports, opposes, or evades are, respectively, 20, 25, and 30; the largest of these, 30, is the maximin value. Similarly, for each strategy it chooses, B determines the maximum percentage of votes A can win (and thus the minimum that B can win). In this case if B supports, opposes, or evades, the maximum A gets is 80, 30, or 80, respectively. B obtains its highest percentage by minimizing A's maximum percentage of the vote. The smallest of A's maximum values is 30, and 30 is therefore B's minimax value. Because both the minimax and the maximin values are 30, 30 is a saddle point. The two parties might as well announce their strategies in advance; neither gains from the knowledge.

 

Mixed strategies; minimax theorem

When saddle points exist, the proper strategies and outcome are clear, but in some games there are no saddle points. The normal form of such a game is given below.

A guard is hired to protect two safes: safe A contains $10,000 and safe B contains $100,000. The guard can protect only one safe from the robber at a time. The robber and the guard must decide in advance, without knowing what the other will do, which safe to approach. If they go to the same safe, the robber gets nothing; if they go to different safes, the robber keeps the contents of the unprotected safe. In such a game, theory does not suggest any one particular strategy; instead, it prescribes that a strategy be chosen in accordance with a probability distribution, which in this simple example is fairly easy to determine. In larger, more complex games it involves solving a problem in linear programming and can be quite difficult.

To calculate the appropriate probability distribution in this case, each player adopts a strategy that makes him indifferent to what his opponent does. It can be assumed that the guard protects safe A with probability p and safe B with probability (1 − p). Thus, if the robber approaches safe A, he is successful whenever the guard protects safe B; in other words, he gets $10,000 with probability (1 − p) and $0 with probability p for an average gain of $10,000 (1 − p). Similarly, if the robber approaches safe B, he gets $100,000 with probability p and $0 with probability (1 − p) for an average gain of $100,000p. The guard will be indifferent to which safe the robber chooses if the average amount the robber gains is the same in both cases—that is, if $10,000 (1 − p) = $100,000p. It can then be calculated that p = 1/11. If the guard protects safe A with probability 1/11 and safe B with probability 10/11, he loses, on the average, no more than $100,000/11, or about $9,091, whatever the robber does. Using the same kind of argument, it can be deduced that the robber will get an average of at least $9,091 if he tries to steal from safe A with probability 10/11 and from safe B with probability 1/11. This solution in terms of mixed strategies is analogous to the solution of the game with a saddle point. The robber and the guard lose nothing if they announce the probabilities with which they will choose their respective strategies; and, if either uses the proper strategy, the (average) outcome will be the one predicted.

The minimax theorem, which von Neumann proved in 1928, states that every finite, two-person zero-sum game has a solution in mixed strategies. Specifically, it states that for every

such game between players A and B there are a value, V, and mixed strategies for players A and B such that, if A adopts his appropriate mixed strategy, the outcome will be at least as favourable to A as V, while, if B adopts his appropriate mixed strategy, the outcome will be no more favourable to A than V. Thus, A and B have the motivation and the power to enforce the outcome V.

 

Utility theory

In the previous examples it was tacitly assumed that the players were trying to maximize their average profits, but in practice players often have other goals. Few people would risk a sure gain of $1,000,000 for an even chance of gaining $10,000,000, for example. In fact, many decisions people make, such as buying insurance policies, playing lottery games, and gambling in a casino, indicate that they are not maximizing their average profits. Game theory does not attempt to indicate what a player's goal should be; instead, it shows the player how to attain his goal, whatever it may be.

Von Neumann and Morgenstern understood this distinction, and so to accommodate all players, whatever their goals, they constructed a theory of utility. They began by listing certain axioms that they felt all “rational” decision makers would follow (for example, if a person likes tea better than milk, and milk better than coffee, then that person should like tea better than coffee). They then proved that for such rational decision makers it was possible to define a utility function that would reflect an individual's preferences; basically, a utility function assigns to each of a player's alternatives a number that conveys the relative attractiveness of that alternative. Maximizing someone's utility automatically determines his most preferred option. In recent years, however, some doubt has been raised about whether people actually behave in accordance with these rational rules.

 

Two-person nonzero-sum games

Most of the early work in game theory involved completely competitive two-person zero-sum games because they are the simplest to treat mathematically. Both players in such games have a clear purpose (to outplay their opponent), and there is general agreement about what constitutes a solution. Most games that arise in practice, however, are nonzero-sum ones; the players have both common and opposed interests. For example, a buyer and a seller are engaged in a nonzero-sum game (the buyer wants a low price and the seller a high one, but both want to make a deal), as are two hostile nations (they may disagree about numerous issues, but both gain if they avoid going to war).

All zero-sum games are the same in one respect: the players are acting at cross-purposes. Nonzero-sum games differ radically from them (and from each other), and many “obvious” properties of zero-sum games are no longer valid. In zero-sum games, for example, players cannot gain (they may or may not lose, but they cannot gain) if they are deprived of some of their strategies. In nonzero-sum games, however, players may well gain if some of their options are no longer available. This might not seem logical at first. One would think that if it benefited a player not to use certain strategies, the player would simply avoid those strategies and choose a more advantageous one, but this is not always possible. For example, in a

region with high unemployment a worker may be willing to accept a low salary rather than lose a job, but, if a minimum wage law makes that option illegal, the worker may be “forced” to require a higher salary than he might otherwise have accepted.

One of the factors that most reveals the difference between zero-sum and nonzero-sum games is the effect of communication on the game. In zero-sum games it never helps a player to give an adversary information, and it never harms a player to learn an opponent's strategy in advance. These rules do not necessarily hold true for nonzero-sum games, however. A player may want his opponent to be well-informed. In a labour–management dispute, for example, if the labour union is prepared for a strike, it behooves it to inform management and thereby possibly achieve its goal without a long, costly conflict. In this example, management is not harmed by the advance information (it, too, benefits by avoiding the costly strike), but in other nonzero-sum games a player can be at a disadvantage if he knows his opponent's strategy. A blackmailer, for example, benefits only if he informs his victim that he will harm the victim unless his terms are met. If he does not give this information to the intended victim, the blackmailer can still do damage but he has no reason to. Thus, knowledge of the blackmailer's strategy works to the victim's disadvantage.

 

Cooperative versus noncooperative games

Communication is irrelevant in zero-sum games because there is no possibility of the players cooperating (their interests are exactly opposite). In nonzero-sum games the ability to communicate, the degree of communication, and even the order in which players communicate can have a profound influence on the outcome of the game. Games in which players are allowed to communicate and make binding agreements are called cooperative, and games in which players are not allowed to communicate are called noncooperative.

In the example shown below, it is to player B's advantage if the game is cooperative and to player A's advantage if the game is noncooperative. (Note that in nonzero-sum games each matrix entry consists of two numbers. Because the combined wealth of the players is not constant, it is not possible to deduce one player's payoff from the payoff of the other; both players' payoffs must be given. Generally, the first number in each entry represents the payoff to the player whose strategies are listed in a column [here, player B], and the second number represents the payoff to the player whose strategies are listed in a row [here, player A].)

Without communication, each player should apply the “sure-thing” principle of maximizing his minimum payoff; that is, each should determine the minimum amount he could expect to get no matter what his opponent does. A would determine that he would do best to choose strategy I no matter what B does (if B chooses i, A gets 2 rather than 1; if B chooses ii, A gets −100 rather than −500). B would similarly determine that he does best to choose i no matter what A does. Using these two strategies, A would gain 2 and B would gain 1. In a cooperative game, however, B can threaten to play ii unless A agrees to play II. If A agrees, his gain is reduced to 1 while B's gain rises to 3; if A does not agree and B carries out his threat, B neither gains nor loses, but A loses 100.

Often, both players may gain from the ability to communicate. Two pilots trying to avoid a midair collision clearly benefit if they are allowed to communicate, and the degree of communication allowed between them may even determine whether or not they crash. Generally, the more two players' interests coincide, the more important and advantageous communication becomes.

The solution to a completely cooperative game in which players share one common goal involves coordinating the players' decisions effectively. It is relatively straightforward, as is the solution to completely competitive, or zero-sum, games. For games in which players have both common and conflicting interests—in other words for most nonzero-sum games, whether cooperative or noncooperative—the solution is much harder to define persuasively.

 

The Nash solution

Although solutions to some nonzero-sum games have been defined in a number of different ways, they sometimes seem inequitable or are not enforceable. One well-known solution to cooperative, two-person games was defined by the mathematician John F. Nash. He assumed that a game had a set of possible outcomes and that associated with each outcome was a utility for each player; from these, Nash selected his solution, a unique outcome that satisfied four conditions. Expressed briefly these were: (1) the solution must be independent of the choice of utility function (if a player prefers x to y and one function assigns x a utility of 10 and y a utility of 1 while a second function assigns them the values 20 and 2, the solution should not change); (2) it must be impossible for both players to simultaneously do better than the Nash solution (a condition known as Pareto optimality); (3) the solution must be independent of irrelevant alternatives (if unattractive options are added to or dropped from the list of alternatives, the Nash solution should not change); and (4) the solution must be symmetrical (if the players reverse their roles, the solution remains the same except that the payoffs are reversed).

The Nash solution seems inequitable in some cases because it is based on a balance of threats (the possibility that no agreement will be reached and the consequent losses to each player) rather than a “fair” outcome. If, for example, a rich man and a poor man are to be given $10,000 provided they can agree on how to share the money (if they fail to agree, they receive nothing), the fair solution would seem to be for each man to take half the total, or $5,000. Following the Nash procedure, however, there must be a utility for each player associated with all possible outcomes. The specific choice of utility functions should not affect the solution (condition 1) as long as each function reflects each man's preferences. In this example it is assumed that the rich man's utility is proportional to money and that the poor man's utility is proportional to the square root of money. This reflects the fact that small amounts of money are precious to the poor man, making him averse to gambling while the rich man is not. In such a case the Nash solution suggests that the threat of reaching no agreement should induce the poor man to accept a third of the $10,000 and give the rich man two-thirds. The Nash solution seeks an outcome in which each player gains the same amount of utility (relative to the nonagreement outcome) and ignores such considerations as past history or the needs of the players.

 

The prisoners' dilemma

To illustrate the kind of difficulties that arise in noncooperative games, one can consider the celebrated prisoners' dilemma (originally formulated by the mathematician Albert W. Tucker.) Two prisoners, A and B, suspected of committing a robbery together, are isolated and urged to confess. Each is concerned only with getting the shortest possible prison sentence for himself, and each must decide whether to confess or remain mute without knowing his partner's decision. Both know the consequences of their decisions: if both confess, both go to jail for five years; if neither confesses, both go to jail for one year (for carrying concealed weapons); and if one confesses while the other does not, the confessor goes free and the silent one goes to jail for 20 years. The normalized form of this game is shown below.

Superficially, the analysis of the prisoners' dilemma is very simple. Each player can apply the “sure-thing” principle. Although A can't be sure what B will do, he knows that he does best to confess when B confesses (he gets five years rather than 20) and also when B remains mute (he serves no time rather than a year); B, of course, will reach the same conclusion. So the solution would seem to be that each prisoner does best to confess and go to jail for five years. Paradoxically, the two robbers would do better if they both adopted the “irrational” strategy of remaining mute; each would then serve only one year in jail. The irony of the prisoners' dilemma is that when each of two (or more) parties acts selfishly and does not cooperate (that is, confess), they do worse than when they act unselfishly and do cooperate (remain silent).

The prisoners' dilemma is not just an intriguing hypothetical problem; real-life situations with similar characteristics have often been observed. For example, two shopkeepers engaged in a price war are involved in a version of the prisoners' dilemma. Each shopkeeper knows that if he has lower prices than his rival, he will attract his rival's customers and increase his own profits. Each therefore decides to lower his prices, with the result that neither gains any customers and both earn smaller profits because they now make less on each sale. Similarly, nations competing in an arms race and farmers increasing crop production can also be seen as practical examples of the prisoners' dilemma (if two nations keep buying more weapons in an attempt to achieve military superiority, neither gains the advantage and both are poorer than when they started; a single farmer can increase his profits by increasing production, but the overproduction resulting when all farmers increase their output causes a market glut and lower profits).

It might seem that the paradox inherent in the prisoners' dilemma could be resolved if the game were played repeatedly. Players would learn that they do best when both act unselfishly and cooperate; if one player failed to cooperate in one game, the other player could retaliate by not cooperating in the next game and both would lose until they began to cooperate again. When the game is repeated a fixed number of times, however, this argument fails. According to the argument, when the two shopkeepers described above set up their stores at a 10-day county fair, each should maintain a high price, knowing that if he does not, his competitor will retaliate the next day. On the 10th day, however, each shopkeeper realizes that his competitor can no longer retaliate (the fair will be closed so there is no next day); therefore each shopkeeper should lower his price on the last day. But if each shopkeeper knows that his rival will lower the price on the 10th day, he has no incentive to maintain the high price on the ninth day. Continuing this reasoning, one concludes that “rational” shopkeepers will have

a price war every day. It is only when the game is played repeatedly and neither player knows when the sequence will end that the cooperative strategy succeeds.

In 1980 the American political scientist Robert Axelrod engaged a number of game-theory experts in a round-robin tournament. In each match two experts competed against one another in a sequence of prisoners' dilemmas. At the start of a match each expert adopted a fixed strategy, which was programmed into a computer. A “nice” strategy was defined as one in which a player always cooperated with a cooperative adversary. If a player's opponent did not cooperate during one play, most strategies prescribed noncooperation on the next one, but a player with a “forgiving” strategy reverted rapidly back to cooperation once his opponent started cooperating again. In this experiment it turned out that every nice strategy outperformed every one that was not nice. Furthermore, of the nice strategies, the forgiving ones performed best.

One fascinating and unexpected application of game theory in general, and the prisoners' dilemma in particular, occurs in biology. When two males confront one another (whether competing for a mate or for some disputed territory), they can behave either like “hawks”—fighting until one is maimed, killed, or flees—or like “doves”—posturing a bit but leaving before any serious harm is done. (The doves, in effect, cooperate while the hawks do not.) Neither type of behaviour is ideal for survival; a species containing only hawks would have a very high casualty rate, and a species consisting exclusively of doves would be vulnerable to an invasion (or mutation) by hawks since the growth rate of the more competitive hawks would be much higher initially than that of the doves. Thus, a species with males consisting exclusively of either hawks or doves would be unstable. The English biologist John Maynard Smith showed that a third type of male behaviour, which he called “bourgeois,” would be more stable than either pure hawk or pure dove. A bourgeois may act like either a hawk or a dove, depending upon some external cue; it may, for example, fight tenaciously when it meets a rival in its own territory but yield when it meets the same rival elsewhere. In effect, bourgeois animals submit their conflict to external arbitration to avoid a prolonged, mutually destructive conflict.

Smith constructed the following table of payoffs in which various possible outcomes (e.g., death, maiming, successful mating), and the costs and benefits associated with them (e.g., cost of lost time), were weighted in terms of the expected number of genes propagated.

Smith showed that a bourgeois invasion would be successful against a completely hawk population by observing that when a hawk confronts a hawk it loses 5, while a bourgeois loses only 2.5. (Because the population is assumed to be predominantly hawk, the success of the invasion can be predicted by comparing the average number of offspring a hawk will have when it confronts another hawk with the number of offspring a bourgeois will have when confronting a hawk.) Clearly, a bourgeois invasion against a completely dove population would be successful as well. On the other hand, a completely bourgeois population could not be invaded by either hawks or doves because bourgeois get 5 against bourgeois, which is more then either hawks or doves get when confronting bourgeois.

Smith gave several examples that showed how the bourgeois strategy is used in practice. The male speckled wood butterfly seeks sunlit spots on the forest floor where females are often found. There is a shortage of such spots, however, and, in a confrontation between a stranger and an inhabitant, after a brief duel (in which the combatants circle one another) the stranger

yields. The dueling skills of the adversaries have little effect on the outcome. If one butterfly is forcibly placed on another's territory so that each considers the other the aggressor, the two butterflies duel with righteous indignation for a much longer time.

 

Mixed strategies

In two-person zero-sum games it is often prudent for players to use mixed strategies in order to hide their intentions. The American social scientist Henry Hamburger showed that mixed strategies are useful in nonzero-sum games as well. He described the dilemma facing a community in which the number of high-speed automobile accidents was rising and the cost of strict enforcement of speed limits was prohibitively high. He assumed that all of the costs and benefits to drivers and to society associated with a speed limit could be calculated, and he then tried to determine how well the speed limit could be enforced. After considering such factors as the cost of strict enforcement, the risk of accidents to the driver and their cost to the community, the penalty to the driver when caught, and the time the driver saved by violating speed laws, he constructed the following payoff matrix.

It can be seen that the cost to the community of strict enforcement is high whatever the driver does. Thus, a driver who tends to speed will deduce that strict enforcement is not likely and will violate the law; the community deduces that it does better not to enforce the law strictly even if it is aware of the motorist's violation. If, however, the community rejects both pure strategies (complete enforcement and complete neglect) and announces that it will enforce the law precisely 10 percent of the time—and actually does so—the matrix is transformed into the one shown below.

A driver who violates the law now loses 10, which should induce him to obey the law (and break even). The cost to the community of this partial enforcement is thus only 2, which is lower than the cost of ignoring the law and allowing drivers to violate it. The community clearly benefits by using the mixed strategy.

 

N-person games

Theoretically, n-person games in which the players are not allowed to communicate are not essentially different from two-person games. If cooperation is allowed, however, there is an opportunity for some players to join forces and act as a single unit, significantly changing the nature of the game. In such games the fate of each player is much more dependent upon the decisions of others than in one- and two-person games; consequently, the concept of a “best” strategy is even harder to define persuasively.

 

The von Neumann–Morgenstern theory

Von Neumann and Morgenstern were the first to construct a theory of n-person games. They assumed that various subgroups of players might join together to form coalitions, each of which has an associated value defined as the minimum amount that the coalition can attain using its own efforts exclusively. (In practice, such subgroups might be blocs in a legislative body or a group of merging businesses.) They described these n-person games in characteristic-function form—that is, by listing first the players, then the coalitions the players can form, and finally the respective values of these coalitions. They also assumed that the characteristic function is superadditive; in other words, the value of a grand coalition of two subcoalitions is at least as great as the sum of the values of the two coalitions that formed it.

Clearly, the sum of the payments to the players in each coalition that forms must equal the value of that coalition, and each player must receive no less than what he could receive playing alone (if he did not, he would not join). Each set of payments to the players describes one possible outcome of an n-person game and is called an imputation. An imputation X is said to dominate another imputation Y with an effective set of players, S, if (1) each player in S gets more with X than with Y and (2) the players in S receive a total payment that does not exceed the coalition value of S. This means that players in S prefer the payoff X to Y and have the power to enforce this preference.

Von Neumann and Morgenstern defined the solution to an n-person game as a set of imputations satisfying two conditions: (1) no imputation in the solution dominates another imputation in the solution and (2) any imputation not in the solution is dominated by another one in the solution. A von Neumann–Morgenstern solution is not a prescription for a single outcome that will or should occur; it is, rather, a set of outcomes, any one of which might occur. It is stable because, for the members of the effective set, S, any imputation outside the solution is dominated by (and therefore less preferable than) an imputation within the solution, and each imputation in the solution is viable because it is not dominated by any other imputation in the solution.

In any given game there are generally many solutions. A simple three-person game that illustrates this fact is one in which any two players can receive one unit (which they may distribute among themselves as they wish). In such a case the value of each two-person coalition (and the three-person coalition as well) is one. One solution to the game consists of three imputations, each with one player receiving nothing and the other two receiving 1/2. There is no self-domination within the solution; if one imputation is substituted for another, one player gets more, one gets less, and one gets the same (for domination, both of the players forming a coalition must gain). In addition, any imputation outside the solution is dominated by one in the solution; for any imputation outside the solution the two players with the lowest payoffs must each get less than 1/2, and so this imputation is dominated by an imputation in the solution in which these two players each get 1/2. If this solution results, then at any given time one of its three imputations will be in effect; von Neumann and Morgenstern do not predict which.

A second solution to this game consists of all the imputations in which player I gets 1/4 and players II and III share the remaining 3/4. Although this solution results in a different set of outcomes than the first solution, it, too, satisfies von Neumann and Morgenstern's two conditions. For any imputation within the solution, player I always gets 1/4 and therefore

cannot gain. In addition, because players II and III share a fixed sum, if one of them gains in a proposed imputation, the other must lose. Thus, no imputation in the solution dominates another imputation in the solution. For any imputation not in the solution, player I must get either more or less than 1/4. If he gets more than 1/4, players II and III share less than 3/4 and therefore can both do better with an imputation within the solution. If player I gets less than 1/4, say 1/8, he always does better with an imputation in the solution. Players II and III now have more to share; but, no matter how they split the new total of 7/8, there is an imputation in the solution that one of them will prefer. If they share equally, each gets 7/16; but player II, for example, can get more in the imputation (1/4, 1/2, 1/4), which is in the solution. If players II and III do not divide the 7/8 equally, the player who gets the smaller amount can always do better with an imputation in the solution. Thus, any imputation outside the solution is dominated by one inside the solution. It can be similarly shown that all of the imputations in which player II gets 1/4 and players I and III share 3/4, as well as the set of all imputations in which player III gets 1/4 and players I and II share 3/4, also constitute solutions to the game.

Although there may be many solutions to a game (each representing a different “standard of behaviour”) it was not clear at first that there would always be at least one. Von Neumann and Morgenstern found no game which could be shown not to have a solution, and they deemed it important that no such game be found. The existence of such a game—a fairly complicated 10-person game discovered in 1967 by the American mathematician William F. Lucas—indicates that the von Neumann–Morgenstern solution is not universally applicable, but the concept remains compelling, especially since no definitive theory of n-person games exists.

 

The Aumann–Maschler theory

The mathematicians Robert J. Aumann and Michael Maschler defined a different type of solution for n-person games. They started with the same basic structure that von Neumann and Morgenstern did—an n-person game in characteristic-function form—and assumed that the players formed some coalition structure. They then tried to determine what the appropriate payoffs should be for the members of each coalition (they did not predict which coalitions would form, however). Each imputation was determined to be inside or outside the solution on its own merits, independently of any other imputations found in the solution.

Specifically, Aumann and Maschler began by considering a coalition structure and a tentative payoff for all the players. They then permitted each player to “object” to other members in his own coalition; the aggrieved member objected by showing that he could form a new coalition elsewhere in which not only did he receive a larger payoff but the members in the new, prospective coalition received a larger payoff as well. The player to whom the objection was directed could “counterobject” by showing that he too could form a new coalition in which both he and his new coalition mates gained at least as much as they did from the original coalition and from the objecting member's proposed coalition.

The basic concepts of the Aumann–Maschler theory are explained in the following example. An agent notifies three actors that there is a job for precisely two of them. The actors vary in talent and experience, so the various pairs will not receive the same compensation; A and B can earn $60, A and C $80, and B and C $100. Before any pair can accept an offer, it must

first decide how the payment will be shared. The first pair to reach an agreement gets the job. This game can be represented by the figure below.

B and C can earn the largest total sum, so it can be assumed that they are at least considering forming a coalition. B and C must then begin negotiating how they will divide the $100. B, for example, might offer to take $45 and give C $55. If C should then object that he could offer A $23 and take $57 for himself, B has no valid reply. In the new coalition, C increases his own payoff as well as A's (who would otherwise get nothing), and if B were to compete for A and offer him $23, he would have only $37 for himself—less than he is currently asking in the B–C coalition. Rather than get nothing, however, B might decide to offer C $58, taking $42 for himself. Negotiations would continue in this manner until the proper payoff (in this case, $40 for B and $60 for C) was being considered. With this payoff, either B or C could object by joining A, but then the other could counterobject by showing that he could offer A the same amount. The Aumann–Maschler theory does not predict which two-person coalition will form (or even that one will form at all); but it shows that if any of the three potential two-person coalitions does form, then the appropriate payments for A, B, and C are $20, $40, and $60, respectively.

The Aumann–Maschler solution gives a particular player a better idea of what he may aspire to, but it is not unique, even for a fixed coalition structure. For example, in a case in which there are four players—two buyers and two sellers—and any buyer and seller together can make $100, it is clear that two coalitions should form, each with one buyer and one seller. The theory, however, does not prescribe how the payoffs will be divided among either pair; it predicts only that each of the two sellers and each of the two buyers will receive the same amount.

The Aumann–Maschler solution, like the Nash solution, is based upon power, not equity. The following problem, for example, has a seemingly unfair Aumann–Maschler solution. An employer and any one of two employees can form a two-person coalition with a value of 100, while the value of the coalition consisting of the two employees is zero. According to Aumann and Maschler, if an employee received more than zero, he would have no recourse if the employer objected and threatened to hire the other employee instead. Thus, the employer must get 100 in any coalition he joins (with more than a single player) while each employee gets nothing. This solution suggests that employees would do well to unite and act as one player, which they effectively do when they bargain collectively, thereby reducing the problem to a two-person game.

Unlike the von Neumann–Morgenstern theory, for any coalition structure there is always at least one Aumann–Maschler solution.

 

Applications

Voting games

In situations involving voting there is a well-defined procedure for decision making by a well-defined set of players, and any outcome is dependent upon these decisions. Moreover, it often behooves a voter to anticipate the votes of others before he votes himself. It is not surprising, therefore, that many of the applications of n-person game theory are concerned with voting.

One of the most basic problems game theory has tried to answer is how to choose a “good” voting procedure. It might seem that finding a suitable voting system is simple, but in fact there are many difficulties, as can be seen in the following example. A voting body has three alternatives—A, B, and C—from which it must choose one. A decision rule picks one of these three alternatives on the basis of how the electorate voted. Voter preference is described by the notation (ABC) = k, which means that k voters prefer A to B and B to C.

One decision rule often used in practice is to choose the plurality favourite—the alternative receiving the largest number of votes. Following this rule, if there are 60 voters, (ABC) = 21, (BCA) = 20, (CBA) = 19, and each voter backs his favourite candidate, then A would win. But almost two-thirds of the electorate consider A the worst choice and, in fact, A would lose a two-way election against either B or C by almost 2 to 1.

To avoid this paradox, a two-stage election can be instituted in which the least-preferred alternative is eliminated in the first round, and the winner is then chosen from the remaining two alternatives in the second round. In this type of election, if (BAC) = (CAB) = 21 and (ABC) = 18, then A would be eliminated in the first round and B would defeat C 39 to 21 in the second round. In a two-way election between A and B, however, A would win by almost two to one. It is clear that two-stage elections may also provide paradoxical results. In fact, it has been proven that any voting system with at least three alternatives will under certain conditions lead to an outcome that most people would consider inappropriate.

One of the basic difficulties in devising a satisfactory voting system is that group preferences are what mathematicians call intransitive. This means that a group may prefer A to B, B to C, and C to A, as is the case when precisely one-third of the electorate has each of the following preferences: (ABC), (BCA), and (CAB). Another difficulty with voting systems is that voters may vote strategically rather than sincerely. It may seem that in a plurality voting system the most popular first choice would win, but it often serves a voter's purpose to back his second choice. If (ABC) = 21, (BAC) = 20, and (CBA) = 19, the 19 (CBA) voters do well to switch from C to B; that way, they get their second, rather than their third, choice. This is commonly known as voting for the “lesser evil.”

 

The Shapley value

When members of a voting body control different numbers of votes, it is natural to ask how the voting powers of these members compare. Game theoreticians have therefore devoted much thought to the problem of how to calculate the “power” of an individual or a coalition. It is intuitively clear that the power of a senator differs from that of the president, but it is

another thing to assign actual quantitative values to these powers. At first, it might seem that power is proportional to the number of votes cast, but the following example demonstrates that this cannot be right. If A, B, and C control three, two, and two votes, respectively, and decisions are made by majority vote, then, clearly, everyone has the same power despite A's extra vote.

The American mathematician Lloyd S. Shapley devised a measure of the power of a coalition based upon certain assumptions (e.g., the power of a coalition depends upon the characteristic function only). In voting games, which are sometimes called “simple games,” every coalition has the value 1 (if it has enough votes to win) or 0 (if it does not). The sum of the powers of all the players is 1. If a player has 0 power, his vote has no influence on the outcome of the vote; and if a player has a power of 1, the outcome depends on his vote only. The key to calculating voting power is determining the frequency with which a player is the swing voter. In other words, it is assumed that the members of a body vote for a measure in every possible permutation (or order). The fraction of all permutations in which someone is the swing voter—that is, where the measure had insufficient votes to pass before the swing vote but enough to pass after it—is defined to be the Shapley value, or power, of that voter.

For example, in an electorate with five members—A, B, C, D, and E—with one, two, three, four, and five votes, respectively, decision is by majority rule. Despite the disparity in the number of votes each member controls, A and B each have a Shapley value of 1/15, C and D each have a value of 7/30, and E has a value of 2/5. This reflects the fact that any winning coalition containing A and not containing B will still win if A and B both change their votes—they each have exactly the same power. A similar statement can be made concerning C and D.

The Shapley value has been used in a number of ways, some of them quite surprising. It has been used to calculate the relative power of the permanent and nonpermanent members in the United Nations Security Council (the five permanent members have 98 percent of the power), the power of a U.S. congressman as compared to that of a senator or the president, and the power of members of various city councils. Using a variation of this concept, John Banzhaf, an American attorney, successfully challenged a weighted system of voting in Nassau County, New York, in which six municipalities had, respectively, nine, nine, seven, three, one, and one members on the Board of Supervisors. Banzhaf proved that the three municipalities with the lowest weights were effectively disenfranchised because two of the top three weights guaranteed 16 out of 30 votes, a majority.

The Shapley value shows that if a voting body has one powerful member with 15 votes and 30 members with one vote each, the power of the strong member is 15/31, which is considerably more than the third of the votes that that member controls. In general, one large bloc of votes amidst a sea of small blocs is usually disproportionately strong. (For this reason, populous states are stronger in the U.S. electoral college than their votes would indicate, even though the fact that every state, whatever the size of its population, has two senators seems to give an edge to the smaller states.) Conversely, when a voting body consists of two large, approximately equal blocs and one small bloc, it is the small bloc that has the disproportionate power. (If there are a total of three voters and two have 100 votes and one has one vote, all have the same power if the majority rules.) It is because of these two phenomena that political conventions are unstable. When two equally powerful factions are vying for power, the uncommitted parties on the sidelines have a great deal of leverage; but as soon as one of the rival factions shows signs of succeeding, its power increases quickly at

the expense of its main competitor and those on the sidelines, leading to the well-known “bandwagon” effect.

Curiously, the Shapley value, which was designed to measure the power of a coalition, has also been applied for an entirely different purpose, as is demonstrated in the following example. Four airplanes share a runway, and, because they vary in size, the length of runway each airplane requires varies as well. If the “value” of a “coalition” of airplanes is defined to be the cost of the shortest runway required to serve the airplanes in that coalition, the Shapley value of each airplane represents its equitable share of the cost of the runway. This method of calculating one party's share of total cost has been adopted by accountants in practice and reported upon in accounting journals.

Various principles and methods of game theory have similarly been applied to fields other than those for which they were originally developed. Although a relatively new area of study, game theory plays a significant role in such diverse subjects as management science, behavioral science, political science, economics, information theory, control theory, and pure mathematics. It has proved especially useful in analyzing the effect of information, solving problems of cost or resource allocation, calculating relative political or voting power, determining equilibrium points, and explaining patterns of observed behaviour.

The seminal work in game theory is John Von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, 3rd ed. (1953, reprinted 1980). Game theory as a whole is covered in the following books, listed in order of increasing difficulty: Anatol Rapoport, Fights, Games, and Debates (1960, reprinted 1967); Morton D. Davis, Game Theory: A Nontechnical Introduction, rev. ed. (1983); R. Duncan Luce and Howard Raiffa, Games and Decisions: Introduction and Critical Survey (1957, reprinted 1967); and Guillermo Owen, Game Theory, 2nd ed. (1982). Applications of game theory are presented in Nesmith C. Ankeny, Poker Strategy: Winning with Game Theory (1981, reprinted 1982); Robert Axelrod, The Evolution of Cooperation (1984), concerned with evolution and ecology; Steven J. Brams, Game Theory and Politics (1975); and Henry Hamburger, Games as Models of Social Phenomena (1979). Useful essays and journal articles include Robert J. Aumann and Michael Maschler, “The Bargaining Set for Cooperative Games,” in M. Dresher, L.S. Shapley, and A.W. Tucker (eds.), Advances in Game Theory (1964), pp. 443–476, the basis of the Aumann–Maschler solution concept; Daniel Kahneman and Amos Tversky, “The Psychology of Preferences,” Scientific American, 246(1):160–173 (January 1982), a discussion of the validity of the assumptions that underlie utility theory; L.S. Shapley, “A Value for N-Person Games,” in H.W. Kuhn and A.W. Tucker (eds.), Contributions to the Theory of Games, vol. 2 (1953), pp. 307–317, the foundation of the Shapley value; John Maynard Smith, “The Evolution of Behavior,” Scientific American, 239(3):176–192 (September 1978); and Philip D. Straffin, Jr., “The Bandwagon Curve,” American Journal of Political Science, 21(4):695–709 (November 1977), which explains “bandwagon” effect on the basis of the Shapley value.

NAME-DEBAYAN BHATTACHARYAY.

CLASS-FYBMS

ROLL NO- 9

DIVISION-A

SUBJECT- ECONOMICS

TOPIC- GAME THEORY.