Table 4branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent. This interdependence causes each player to consider the other player’s possible decisions, or strategies, in formulating his own strategy. A solution to a game describes the optimal decisions of the players, who may have similar, opposed, or mixed interests, and the outcomes that may result from these decisions.
Although game theory can be and has been used to analyze parlour games, its applications are much broader. In fact, game theory was originally developed by the Hungarian-born American mathematician John von Neumann and his Princeton University colleague Oskar Morgenstern, a German-born American economist, to solve problems in economics. In their book The Theory of Games and Economic Behavior (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, wherein players anticipate each other’s moves, and therefore requires a new kind of mathematics, which they called game theory. (The name may be somewhat of a misnomer—game theory generally does not share the fun or frivolity associated with games.)
Game theory has been applied to a wide variety of situations in which the choices of players interact to affect the outcome. In stressing the strategic aspects of decision making, or aspects controlled by the players rather than by pure chance, the theory both supplements and goes beyond the classical theory of probability. It has been used, for example, to determine what political coalitions or business conglomerates are likely to form, the optimal price at which to sell products or services in the face of competition, the power of a voter or a bloc of voters, whom to select for a jury, the best site for a manufacturing plant, and the behaviour of certain animals and plants in their struggle for survival. It has even been used to challenge the legality of certain voting systems.
It would be surprising if any one theory could address such an enormous range of “games,” and in fact there is no single game theory. A number of theories have been proposed, each applicable to different situations and each with its own concepts of what constitutes a solution. This article describes some simple games, discusses different theories, and outlines principles underlying game theory. Additional concepts and methods that can be used to analyze and solve decision problems are treated in the article optimization.
Games can be classified according to certain significant features, the most obvious of which is the number of players. Thus, a game can be designated as being a one-person, two-person, or n-person (with n greater than two) game, with games in each category having their own distinctive features. In addition, a player need not be an individual; it may be a nation, a corporation, or a team comprising many people with shared interests.
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 all of their opponents’ cards.
The extent to which the goals of the players coincide or conflict is another basis for classifying games. Constant-sum games are games of total conflict, which are also called games of pure competition. Poker, for example, is a constant-sum game because the combined wealth of the players remains constant, though its distribution shifts in the course of play.
Players in constant-sum games have completely opposed interests, whereas in variable-sum games they may all be winners or losers. In a labour-management dispute, for example, the two parties certainly have some conflicting interests, but both will benefit if a strike is averted.
Variable-sum games can be further distinguished as being either cooperative or noncooperative. In cooperative games players can communicate and, most important, make binding agreements; in noncooperative games players may communicate, but they cannot make binding agreements, such as an enforceable contract. An automobile salesperson and a potential customer will be engaged in a cooperative game if they agree on a price and sign a contract. However, the dickering that they do to reach this point will be noncooperative. Similarly, when people bid independently at an auction they are playing a noncooperative game, even though the high bidder agrees to complete the purchase.
Finally, a game is said to be finite when each player has a finite number of options, the number of players is finite, and the game cannot go on indefinitely. Chess, checkers, poker, and most parlour games are finite. Infinite games are more subtle and will only be touched upon in this article.
A game can be described in one of three ways: in extensive, normal, or characteristic-function form. (Sometimes these forms are combined, as described in the section Theory of moves.) Most parlour games, which progress step by step, one move at a time, can be modeled as games in extensive form. Extensive-form games can be described by a “game tree,” in which each turn is a vertex of the tree, with each branch indicating the players’ successive choices.
The normal (strategic) form is primarily used to describe two-person games. In this form a game is represented by a payoff matrix, wherein each row describes the strategy of one player and each column describes the strategy of the other player. The matrix entry at the intersection of each row and column gives the outcome of each player choosing the corresponding strategy. The payoffs to each player associated with this outcome are the basis for determining whether the strategies are “in equilibrium,” or stable.
The characteristic-function form is generally used to analyze games with more than two players. It indicates the minimum value that each coalition of players—including single-player coalitions—can guarantee for itself when playing against a coalition made up of all the other players.
One-person games are also known as games against nature. With no opponents, the player only needs to list available options and then choose the optimal outcome. When chance is involved the game might seem to be more complicated, but in principle the decision is still relatively simple. For example, a person deciding whether to carry an umbrella weighs the costs and benefits of carrying or not carrying it. While this person may make the wrong decision, there does not exist a conscious opponent. That is, nature is presumed to be completely indifferent to the player’s decision, and the person can base his decision on simple probabilities. One-person games hold little interest for game theorists.
The simplest game of any real theoretical interest is a two-person constant-sum game of perfect information. Examples of such games include chess, checkers, and the Japanese game of go. In 1912 the German mathematician Ernst Zermelo proved that such games are strictly determined; by making use of all available information, the players can deduce strategies that are optimal, which makes the outcome preordained (strictly determined). In chess, for example, exactly one of three outcomes must occur if the players make optimal choices: (1) White wins (has a strategy that wins against any strategy of Black); (2) Black wins; or (3) White and Black draw. In principle, a sufficiently powerful supercomputer could determine which of the three outcomes will occur. However, considering that there are some 1043 distinct 40-move games of chess possible, there seems no possibility that such a computer will be developed now or in the foreseeable future. Therefore, while chess is of only minor interest in game theory, it is likely to remain a game of enduring intellectual interest.
A “saddlepoint” in a two-person constant-sum game is the outcome that rational players would choose. (Its name derives from its being the minimum of a row that is also the maximum of a column in a payoff matrix—to be illustrated shortly—which corresponds to the shape of a saddle.) A saddlepoint always exists in games of perfect information but may or may not exist in games of imperfect information. By choosing a strategy associated with this outcome, each player obtains an amount at least equal to his payoff at that outcome, no matter what the other player does. This payoff is called the value of the game; as in perfect-information games, it is preordained by the players’ choices of strategies associated with the saddlepoint, making such games strictly determined.
The normal-form game in Table 1 is used to illustrate the calculation of a saddlepoint. Two political parties, A and B, must each decide how to handle a controversial issue in a certain election. Each party can either support the issue, oppose it, or evade it by being ambiguous. The decisions by A and B on this issue determine the percentage of the vote that each party receives. The entries in the payoff matrix represent party A’s percentage of the vote (the remaining percentage goes to B). When, for example, A supports the issue and B evades it, A gets 80 percent and B 20 percent of the vote.
Assume that each party wants to maximize its vote. A’s decision seems difficult at first because it depends on B’s choice of strategy. A does best to support if B evades, oppose if B supports, and evade if B opposes. A must therefore consider B’s decision before making its own. Note that no matter what A does, B obtains the largest percentage of the vote (smallest percentage for A) by opposing the issue rather than supporting it or evading it. Once A recognizes this, its strategy obviously should be to evade, settling for 30 percent of the vote. Thus, a 30 to 70 percent division of the vote, to A and B respectively, is the game’s saddlepoint.
A more systematic way of finding a saddlepoint is to determine the so-called maximin and minimax values. A first determines the minimum percentage of votes it can obtain for each of its strategies; it then finds the maximum of these three minimum values, giving the maximin. 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 B chooses, it determines the maximum percentage of votes A will win (and thus the minimum that it can win). In this case, if B supports, opposes, or evades, the maximum A will get is 80, 30, and 80, respectively. B will obtain its largest percentage by minimizing A’s maximum percent of the vote, giving the minimax. The smallest of A’s maximum values is 30, so 30 is B’s minimax value. Because both the minimax and the maximin values coincide, 30 is a saddlepoint. The two parties might as well announce their strategies in advance, because the other party cannot gain from this knowledge.
When saddlepoints exist, the optimal strategies and outcomes can be easily determined, as was just illustrated. However, when there is no saddlepoint the calculation is more elaborate, as illustrated in Table 2.
A guard is hired to protect two safes in separate locations: S1 contains $10,000 and S2 contains $100,000. The guard can protect only one safe at a time from a safecracker. The safecracker and the guard must decide in advance, without knowing what the other party will do, which safe to try to rob and which safe to protect. When they go to the same safe, the safecracker gets nothing; when they go to different safes, the safecracker gets the contents of the unprotected safe.
In such a game, game theory does not indicate that any one particular strategy is best. Instead, it prescribes that a strategy be chosen in accordance with a probability distribution, which in this simple example is quite easy to calculate. In larger and more complex games, finding this strategy involves solving a problem in linear programming, which can be considerably more difficult.
To calculate the appropriate probability distribution in this example, each player adopts a strategy that makes him indifferent to what his opponent does. Assume that the guard protects S1 with probability p and S2 with probability 1 − p. Thus, if the safecracker tries S1, he will be successful whenever the guard protects S2. In other words, he will get $10,000 with probability 1 − p and $0 with probability p for an average gain of $10,000(1 − p). Similarly, if the safecracker tries S2, he will get $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 safecracker chooses if the average amount stolen is the same in both cases—that is, if $10,000(1 − p) = $100,000p. Solving for p gives p = 1/11. If the guard protects S1 with probability 1/11 and S2 with probability 10/11, he will lose, on average, no more than about $9,091 whatever the safecracker does.
Using the same kind of argument, it can be shown that the safecracker will get an average of at least $9,091 if he tries to steal from S1 with probability 10/11 and from S2 with probability 1/11. This solution in terms of mixed strategies, which are assumed to be chosen at random with the indicated probabilities, is analogous to the solution of the game with a saddlepoint (in which a pure, or single best, strategy exists for each player).
The safecracker and the guard give away nothing if they announce the probabilities with which they will randomly choose their respective strategies. On the other hand, if they make themselves predictable by exhibiting any kind of pattern in their choices, this information can be exploited by the other player.
The minimax theorem, which von Neumann proved in 1928, states that every finite, two-person constant-sum game has a solution in pure or mixed strategies. Specifically, it says that for every such game between players A and B, there is a value v and strategies for A and B such that, if A adopts its optimal (maximin) strategy, the outcome will be at least as favourable to A as v; if B adopts its optimal (minimax) strategy, the outcome will be no more favourable to A than v. Thus, A and B have both the incentive and the ability to enforce an outcome that gives an (expected) payoff of v.
In the previous example it was tacitly assumed that the players were maximizing their average profits, but in practice players may consider other factors. For example, few people would risk a sure gain of $1,000,000 for an even chance of winning either $3,000,000 or $0, even though the expected (average) gain from this bet is $1,500,000. In fact, many decisions that people make, such as buying insurance policies, playing lotteries, and gambling at a casino, indicate that they are not maximizing their average profits. Game theory does not attempt to state what a player’s goal should be; instead, it shows how a player can best achieve his goal, whatever that goal is.
Von Neumann and Morgenstern understood this distinction; to accommodate all players, whatever their goals, they constructed a theory of utility. They began by listing certain axioms that they thought all rational decision makers would follow (for example, if a person likes tea better than coffee, and coffee better than milk, then that person should like tea better than milk). They then proved that it was possible to define a utility function for such decision makers that would reflect their preferences. In essence, a utility function assigns a number to each player’s alternatives to convey their relative attractiveness. Maximizing someone’s expected utility automatically determines a player’s most preferred option. In recent years, however, some doubt has been raised about whether people actually behave in accordance with these axioms, and alternative axioms have been proposed.
Much of the early work in game theory was on two-person constant-sum games because they are the easiest to treat mathematically. The players in such games have diametrically opposed interests, and there is a consensus about what constitutes a solution (as given by the minimax theorem). Most games that arise in practice, however, are variable-sum games; the players have both common and opposed interests. For example, a buyer and a seller are engaged in a variable-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).
Some “obvious” properties of two-person constant-sum games are not valid in variable-sum games. In constant-sum games, for example, both players cannot gain (they may or may not lose, but they cannot both gain) if they are deprived of some of their strategies. In variable-sum games, however, players may gain if some of their strategies are no longer available. This might not seem possible at first. One would think that if a player benefited from not using certain strategies, the player would simply avoid those strategies and choose more advantageous ones, but this is not always the case. For example, in a region with high unemployment a worker may be willing to accept a lower salary to obtain or keep a job, but if a minimum wage law makes that option illegal, the worker may be “forced” to accept a higher salary.
The effect of communication is particularly revealing of the difference between constant-sum and variable-sum games. In constant-sum games it never helps a player to give an adversary information, and it never hurts a player to learn an opponent’s optimal strategy (pure or mixed) in advance. However, these properties do not necessarily hold in variable-sum games. Indeed, a player may want an opponent to be well-informed. In a labour-management dispute, for example, if the labour union is prepared to strike, it behooves the union to inform management and thereby possibly achieve its goal without a strike. In this example, management is not harmed by the advance information (it, too, benefits by avoiding a costly strike). In other variable-sum games, knowing an opponent’s strategy can sometimes be disadvantageous. For example, a blackmailer can only benefit if he first informs his victim that he will harm him—generally by disclosing some sensitive and secret details of the victim’s life—if his terms are not met. For such a threat to be credible, the victim must fear the disclosure and believe that the blackmailer is capable of executing the threat. (The credibility of threats is a question that game theory studies.) Although a blackmailer may be able to harm a victim without any communication taking place, a blackmailer cannot extort a victim unless he first adequately informs the victim of his intent and its consequences. Thus, the victim’s knowledge of the blackmailer’s strategy, including his ability and will to carry out the threat, works to the blackmailer’s advantage.
Another approach to inducing cooperation in PD and other variable-sum games is the theory of moves (TOM). Proposed by the American political scientist Steven J. Brams, TOM allows players, starting at any outcome in a payoff matrix, to move and countermove within the matrix, thereby capturing the changing strategic nature of games as they evolve over time. In particular, TOM assumes that players think ahead about the consequences of all of the participants’ moves and countermoves when formulating plans. Thereby, TOM embeds extensive-form calculations within the normal form, deriving advantages of both forms: the nonmyopic thinking of the extensive form disciplined by the economy of the normal form.
To illustrate the nonmyopic perspective of TOM, consider what happens in PD as a function of where play starts:
Such rational moves are not beyond the pale of most players. Indeed, they are frequently made by those who look beyond the immediate consequences of their own choices. Such far-sighted players can escape the dilemma in PD—as well as poor outcomes in other variable-sum games—provided play does not begin noncooperatively. Hence, TOM does not predict unconditional cooperation in PD but, instead, makes it a function of the starting point of play.
One fascinating and unexpected application of game theory in general, and PD in particular, occurs in biology. When two males confront each other, 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. (In effect, the doves cooperate while the hawks do not.) Neither type of behaviour, it turns out, is ideal for survival: a species containing only hawks would have a high casualty rate; a species containing only doves would be vulnerable to an invasion by hawks or a mutation that produces hawks, because the population growth rate of the competitive hawks would be much higher initially than that of the doves.
Thus, a species with males consisting exclusively of either hawks or doves is vulnerable. The English biologist John Maynard Smith showed that a third type of male behaviour, which he called “bourgeois,” would be more stable than that of either pure hawks or pure doves. A bourgeois may act like either a hawk or a dove, depending on some external cues; for example, it may 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 and mutually destructive struggle.
As shown in Table 5, Smith constructed a payoff matrix 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, whereas 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 produce when it confronts another hawk with the average number of offspring a bourgeois will produce when confronting a hawk.) Patently, a bourgeois invasion against a completely dove population would be successful as well, gaining the bourgeois 6 offspring. On the other hand, a completely bourgeois population cannot be invaded by either hawks or doves, because the bourgeois gets 5 against bourgeois, which is more than either hawks or doves get when confronting bourgeois. Note in this application that the question is not what strategy a rational player will choose—animals are not assumed to make conscious choices, though their types may change through mutation—but what combinations of types are stable and hence likely to evolve.
Smith gave several examples that showed how the bourgeois strategy is used in practice. For example, male speckled wood butterflies seek 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, the stranger yields after a brief duel in which the combatants circle one another. The dueling skills of the adversaries have little effect on the outcome. When 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.
Theoretically, n-person games in which the players are not allowed to communicate and make binding agreements are not fundamentally different from two-person noncooperative games. In the two examples that follow, each involving three players, one looks for Nash equilibria—that is, stable outcomes from which no player would normally depart because to do so would be disadvantageous.
As an example of an n-person noncooperative game, imagine three players, A, B, and C, situated at the corners of an equilateral triangle. They engage in a truel, or three-person duel, in which each player has a gun with one bullet. Assume that each player is a perfect shot and can kill one other player at any time. There is no fixed order of play, but any shooting that occurs is sequential: no player fires at the same time as any other. Consequently, if a bullet is fired, the results are known to all players before another bullet is fired.
Suppose that the players order their goals as follows: (1) survive alone, (2) survive with one opponent, (3) survive with both opponents, (4) not survive, with no opponents alive, (5) not survive, with one opponent alive, and (6) not survive, with both opponents alive. Thus, surviving alone is best, dying alone is worst.
If a player can either fire or not fire at another player, who, if anybody, will shoot whom? It is not difficult to see that outcome (3), in which nobody shoots, is the unique Nash equilibrium—any player that departs from not shooting does worse. Suppose, on the contrary, that A shoots B, hoping for A’s outcome (2), whereby he and C survive. Now, however, C can shoot a disarmed A, thereby leaving himself as the sole survivor, or outcome (1). As this is A’s penultimate outcome (5), in which A and one opponent (B) are killed while the other opponent (C) lives, A should not fire the first shot; the same reasoning applies to the other two players. Consequently, nobody will shoot, resulting in outcome (3), in which all three players survive.
Now consider whether any of the players can do better through collusion. Specifically, assume that A and B agree not to shoot each other; if either shoots another player, they agree it would be C. Nevertheless, if A shoots C (for instance), B could now repudiate the agreement with impunity and shoot A, thereby becoming the sole survivor.
Thus, thinking ahead about the unpleasant consequences of shooting first or colluding with another player to do so, nobody will shoot or collude. Thereby all players will survive if the players must act in sequence, giving outcome (3). Because no player can do better by shooting, or saying they will do so to another, these strategies yield a Nash equilibrium.
Next, suppose that the players act simultaneously; hence, they must decide in ignorance of each others’ intended actions. This situation is common in life: people often must act before they find out what others are doing. In a simultaneous truel there are three possibilities, depending on the number of rounds and whether or not this number is known:
Many applications of n-person game theory are concerned with voting, in which strategic calculations are often rampant. Surprisingly, these calculations can result in the ostensibly most powerful player in a voting body being hurt. For example, assume the chair of a voting body, while not having more votes than other members, can break ties. This would seem to make the chair more powerful, but it turns out that the possession of a tie-breaking vote may backfire, putting the chair at a disadvantage relative to the other members. In this manner the greater resources that a player has may not always translate into greater power, which here will mean the ability of a player to obtain a preferred outcome.
In the three-person noncooperative voting game to be analyzed, players are assumed to rank the possible outcomes that can occur. The problem in finding a solution is not a lack of Nash equilibria, but too many. So the question becomes, Which, if any, are likely to be selected by the players? Specifically, is one more appealing than the others? The answer is “yes,” but it requires extending the idea of a sure-thing strategy to its successive application in different stages of play.
To illustrate the chair’s problem, suppose there are three voters (X, Y, and Z) and three voting alternatives (x, y, and z). Assume that voter X prefers x to y and y to z, indicated by xyz; voter Y’s preference is yzx, and voter Z’s is zxy. These preferences give rise to what is known as a Condorcet voting paradox because the social ordering, according to majority rule, is intransitive: although a majority of voters (X and Z) prefers x to y, and a majority (X and Y) prefers y to z, a majority (Y and Z) also prefers z to x. (The French Enlightenment philosopher Marie-Jean-Antoine-Nicolas Condorcet first examined such voting paradoxes following the French Revolution.) So there is no Condorcet winner—that is, an alternative that would beat every other choice in separate pairwise contests.
Assume that a simple plurality determines the winning alternative. Furthermore, in the event of a three-way tie (there can never be a two-way tie if there are three votes), assume that the chair, X, can break the tie, giving the chair what would appear to be an edge over the other two voters, Y and Z, who have the same one vote but no tie-breaker.
Under sincere voting, everyone votes for his first choice, without taking into account what the other voters might do. In this case, voter X will get his first choice (x) by being able to break a three-way tie in favour of x. However, X’s apparent advantage will disappear if voting is “sophisticated.”
To see why, first note that X has a sure-thing, or dominant, strategy of “vote for x”; it is never worse and sometimes better than any other strategy, whatever the other two voters do. Thus, if the other two voters vote for the same alternative, x will win; and X cannot do better than vote sincerely for x, so voting sincerely is never worse. On the other hand, if the other two voters disagree, X’s tie-breaking vote (along with his regular vote) will be decisive in x’s selection, which is X’s best outcome.
Given the dominant-strategy choice of x on the part of X, then Y and Z face reduced strategy choices, as shown in Table 6 for the first reduction. (It is a reduction because X’s strategy of voting for x is taken as a given.) In this reduction, Y has one, and Z has two, dominated strategies (indicated by D), which are never better and sometimes worse than some other strategy, whatever the other two voters do. For example, observe that “vote for x” by Y always leads to his worst outcome, x. This leaves Y with two undominated strategies, “vote for y” and “vote for z,” which are neither dominant nor dominated strategies: “vote for y” is better than “vote for z” if Z chooses y (leading to y rather than x), whereas the reverse is the case if Z chooses z (leading to z rather than x). By contrast, Z has a dominant strategy of “vote for z,” which leads to outcomes at least as good as and sometimes better than his other two strategies.
When voters have complete information about each other’s preferences, they will eliminate the dominated strategies in the first reduction. The elimination of these strategies gives the second reduction matrix, as shown in Table 7. Then Y, choosing between “vote for y” and “vote for z” in this matrix, would eliminate the now dominated “vote for y” because that choice would result in x’s winning due to the chair’s tie-breaking vote. Instead, Y would choose “vote for z,” ensuring z’s election, which is the next-best outcome for Y. In this manner z, which is not the first choice of a majority and could in fact be beaten by y in a pairwise contest, becomes the sophisticated outcome, which is the outcome produced by the successive elimination of dominated strategies by the voters (beginning with X’s sincere choice of x).
Sophisticated voting results in a Nash equilibrium because none of the players can do better by departing from their sophisticated strategy. This is clearly true for X, because x is his dominant strategy; given X’s choice of x, z is dominant for Z; and given these choices by X and Z, z is dominant for Y. These “contingent” dominance relations, in general, make sophisticated strategies a Nash equilibrium.
Observe, however, that there are four other Nash equilibria in this game. First, the choice of each of x, y, or z by all three voters are all Nash equilibria, because no single voter’s departure can change the outcome to a different one, much less a better one, for that player. In addition, the choice of x by X, y by Y, and x by Z—resulting in x—is also a Nash equilibrium, because no voter’s departure would lead to his obtaining a better outcome.
In game-theoretic terms, sophisticated voting produces a different and smaller game in which some formerly undominated strategies in the larger game become dominated in the smaller game. The removal of such strategies—sometimes in several successive stages—can enable each voter to determine what outcomes are likely. In particular, sophisticated voters can foreclose the possibility that their worst outcomes will be chosen by successively removing dominated strategies, given the presumption that other voters will do likewise.
How does sophisticated voting affect the chair’s presumed extra voting power? Observe that the chair’s tie-breaking vote is not only not helpful but positively harmful: it guarantees that X’s worst outcome (z) will be chosen if voting is sophisticated. When voters’ preferences are not so conflictual (note that the three voters have different first, second, and third choices when, as here, there is a Condorcet voting paradox), the paradox of the chair’s position does not occur, making this paradox the exception rather than the rule.
Von Neumann and Morgenstern were the first to construct a cooperative theory of n-person games. They assumed that various groups of players might join together to form coalitions, each of which has an associated value defined as the minimum amount that the coalition can ensure by its own efforts. (In practice, such groups might be blocs in a legislative body or business partners in a conglomerate.) They described these n-person games in characteristic-function form—that is, by listing the individual players (one-person coalitions), all possible coalitions of two or more players, and the values that each of these coalitions could ensure if a counter-coalition comprising all other players acted to minimize the amount that the coalition can obtain. They also assumed that the characteristic function is superadditive: the value of a coalition of two formerly separate coalitions is at least as great as the sum of the separate values of the two coalitions.
The sum of payments to the players in each coalition must equal the value of that coalition. Moreover, each player in a coalition must receive no less than what he could obtain playing alone; otherwise, he would not join the coalition. Each set of payments to the players describes one possible outcome of an n-person cooperative game and is called an imputation. Within a coalition S, an imputation X is said to dominate another imputation Y if each player in S gets more with X than with Y and if the players in S receive a total payment that does not exceed the coalition value of S. This means that players in the coalition prefer the payoff X to the payoff 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 single outcome but, rather, a set of outcomes, any one of which may occur. It is stable because, for the members of the coalition, any imputation outside the solution is dominated by—and is therefore less attractive than—an imputation within the solution. The imputations within the solution are viable because they are not dominated by any other imputations in the solution.
In any given cooperative game there are generally many—sometimes infinitely many—solutions. A simple three-person game that illustrates this fact is one in which any two players, as well as all three players, receive one unit, which they can divide between or among themselves in any way that they wish; individual players receive nothing. In such a case the value of each two-person coalition, and the three-person coalition as well, is 1.
One solution to this game consists of three imputations, in each of which one player receives 0 and the other two players receive 1/2 each. There is no self-domination within the solution, because if one imputation is substituted for another, one player gets more, one gets less, and one gets the same (for domination, each of the players forming a coalition must gain). In addition, any imputation outside the solution is dominated by one in the solution, because the two players with the lowest payoffs must each get less than 1/2; clearly, this imputation is dominated by an imputation in the solution in which these two players each get 1/2. According to this solution, at any given time one of its three imputations will occur, but von Neumann and Morgenstern do not predict which one.
A second solution to this game consists of all the imputations in which player A receives 1/4 and players B and C share the remaining 3/4. Although this solution gives a different set of outcomes from the first solution, it, too, satisfies von Neumann and Morgenstern’s two conditions. For any imputation within the solution, player A always gets 1/4 and therefore cannot gain. In addition, because players B and C 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 A must get either more or less than 1/4. When A gets more than 1/4, players B and C share less than 3/4 and, therefore, can do better with an imputation within the solution. When player A gets less than 1/4, say 1/8, he always does better with an imputation in the solution. Players B and C 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. When they share equally, each gets 7/16; but player B, for example, can get more in the imputation (1/4, 1/2, 1/4), which is in the solution. When players B and C 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. Similarly, it can be shown that all of the imputations in which player B gets 1/4 and players A and C share 3/4, as well as the set of all imputations in which player C gets 1/4 and players A and B share 3/4, also constitute a solution to the game.
Although there may be many solutions to a game (each representing a different “standard of behaviour”), it was not apparent at first that there would always be at least one in every cooperative game. Von Neumann and Morgenstern found no game without a solution, and they deemed it important that no such game exists. However, in 1967 a fairly complicated 10-person game was discovered by the American mathematician William F. Lucas that did not have a solution. This and later counterexamples indicated that the von Neumann–Morgenstern solution is not universally applicable, but it remains compelling, especially since no definitive theory of n-person cooperative games exists.
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a branch of mathematics used in a variety of disciplines, including economics, military strategy, politics, and other fields, to analyze competitive situations in which outcomes are partly determined by chance; players try to anticipate choices of opponents in order to determine their own best advantage; idea created by John von Neumann and Oskar Morgenstern in their book ’Theory of Games and Economic Behavior’ (1944).
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