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Game Theory and Strategy

Game Theory and Strategy

Philip D. Straffin
Volume: 36
Copyright Date: 1993
Edition: 1
Pages: 256
  • Cite this Item
  • Book Info
    Game Theory and Strategy
    Book Description:

    This important addition to the New Mathematical Library series pays careful attention to applications of game theory in a wide variety of disciplines. The applications are treated in considerable depth. The book assumes only high school algebra, yet gently builds to mathematical thinking of some sophistication. Game Theory and Strategy might serve as an introduction to both axiomatic mathematical thinking and the fundamental process of mathematical modelling. It gives insight into both the nature of pure mathematics, and the way in which mathematics can be applied to real problems.

    eISBN: 978-0-88385-950-6
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-x)
    Philip Straffin
  4. Part I Two-Person Zero-Sum Games

    • 1. The Nature of Games
      (pp. 3-6)

      Game theory is the logical analysis of situations of conflict and cooperation. More specifically, agameis defined to be any situation in which

      i) There are at least twoplayers.A player may be an individual, but it may also be a more general entity like a company, a nation, or even a biological species.

      ii) Each player has a number of possiblestrategies,courses of action which he or she may choose to follow.

      iii) The strategies chosen by each player determine theoutcomeof the game.

      iv) Associated to each possible outcome of the game is a...

    • 2. Matrix Games: Dominance and Saddle Points
      (pp. 7-12)

      We saw in Chapter 1 that a two-person zero-sum game where Rose hasmstrategies and Colin hasnstrategies can be represented by anm × narray of numbers, giving the payoffs from Colin to Rose for each of them · npossible outcomes. Such an array is called anm × nmatrix, so these games are also known asmatrix games.Rose wishes to choose a row of the matrix which will result in a large number; Colin wishes to choose a column which will result in a small number.

      Before reading further, I would...

    • 3. Matrix Games: Mixed Strategies
      (pp. 13-22)

      We saw in Chapter 2 that in some matrix games the row maximin and the column minimax are different numbers, and in those games there is no saddle point. For example, consider

      Since there is no saddle point in this game, neither player would want to play a single strategy with certainty, for the other player could take advantage of such a choice. The only sensible plan is to use some random device to decide which strategy to play. For example, Colin might flip a coin to decide between A and Colin B. Such a plan, which involves playing a...

    • 4. Application to Anthropology: Jamaican Fishing
      (pp. 23-26)

      One important school of anthropological thought, known asfunctionalism,holds that customs, institutions or behavior patterns in a society can be interpreted as functional responses to problems which the society faces. One method, then, of understanding the organization of societies would be to identify problems and stresses, see what kinds of behavior would provide good solutions, and compare a society's behavior patterns to those solutions. For example, incest taboos can be interpreted as societal solutions to genetic problems caused by inbreeding.

      In the 1950’s some pioneering anthropologists began to use game-theoretic ideas in the service of functionalism. For example, Moore...

    • 5. Application to Warfare: Guerrillas, Police, and Missiles
      (pp. 27-31)

      Zero-sum games represent conflict situations, and our solution theory for them prescribes rational strategies for conflict. Since the most extreme form of conflict is war, it is not surprising that some of the first proposed applications of game theory were to tactics in war. Haywood [1954] and Beresford and Peston [1955] describe some applications of game theory to situations from World War II. In this chapter we will consider two applications in more modern settings. Both will be far too simplified to be realistic, but they may give the flavor of what kinds of contributions zero-sum game theory might make...

    • 6. Application to Philosophy: Newcomb’s Problem and Free Will
      (pp. 32-36)

      One of the most persistent problems of philosophy is the problem offree will.Is human will free, or are our actions determined? One way to approach this problem is to consider the possibility of predicting any person’s decision in some unconstrained choice situation. Suppose I ask you to decide consciously to hold up your left hand or your right hand, and tell you that I have written on a slip of paper my prediction of what you will do. Is it possible, in principle, for me to know enough about you to make my prediction with better than chance...

    • 7. Game Trees
      (pp. 37-43)

      In matrix games we have assumed that the players make their choice of strategy simultaneously, without knowledge of what the other player is choosing. This would seem to be a major limitation of the theory, since in real conflict situations decisions are often made sequentially, with information about previous choices becoming available to the players as the situation develops. In this chapter we will consider a method of modeling such sequential choice situations by agame tree.We will find that, perhaps surprisingly, this new model can always be reduced to our old model of a matrix game.

      As a...

    • 8. Application to Business: Competitive Decision Making
      (pp. 44-48)

      In the business world, companies often have to make decisions which involve strategic uncertainty about what other companies will do. Such decisions usually also involve uncertainty about future economic conditions, market size, costs, and other variables. In other words, companies often play games which involve both other players and chance. In games like this the role of information, both about what other companies might do and what chance might do, can be very important.

      In this chapter we will consider a simple example of a competitive situation which can be formulated as a two-person zero-sum game. We will focus specifically...

    • 9. Utility Theory
      (pp. 49-55)

      In the discussion so far, we have mostly assumed that the numerical payoffs in our game matrices or game trees are given. We have not paid much attention to where the numbers come from or exactly what they mean. It is time to consider more thoroughly the process of assigning numbers to outcomes, for the applicability of game theory to real situations rests on the assumption that this can be done in a reasonable way. Von Neumann and Morgenstern were very conscious of this dependence, and they began theTheory of Games and Economic Behaviorby laying the groundwork for...

    • 10. Games Against Nature
      (pp. 56-62)

      In our discussion of Jamaican fishing in Chapter 4, we noted that when one player in a two-person zero-sum game is not a reasoning entity capable of the forethought and adaptive play which game theory assumes of players, the minimax solution concept may not apply. On the other hand, we argued that it may still be applicable, depending on the goals of the rational player in the game. In this chapter we will examine in greater detail possible ways of playing a game against Nature, an unreasoning entity whose strategic choice affects your payoff, but which has no awareness of,...

  5. Part II Two-Person Non-Zero-Sum Games

    • 11. Nash Equilibria and Non-Cooperative Solutions
      (pp. 65-72)

      If a two-person game is not zero-sum, we must write both players’ payoffs to describe the game. If we have a game in which the payoffs to the players do not add to zero, recall from Chapter 9 that the game still might be equivalent to a zero-sum game, in the sense that it could be made zero-sum by a change of utility scales. In such a game, the interests of the two players are strictly opposed, and we can analyze it by zero-sum methods. In general, however, the interests of players in a non-zero-sum game are not strictly opposed,...

    • 12. The Prisoner’s Dilemma
      (pp. 73-80)

      In 1950 Melvin Dresner and Merrill Flood at the RAND Corporation devised Game 12.1 to illustrate that a non-zero-sum game could have an equilibrium outcome which is unique, but fails to be Pareto optimal.

      Later, when presenting this example at a seminar at Stanford University, Albert W. Tucker told a story to go with the game (see [Straffin, 1980]). The players are two prisoners, arrested for a joint crime, who are being interrogated in separate rooms. The clever district attorney tells each one that

      if one of them confesses and the other does not, the confessor will get a reward...

    • 13. Application to Social Psychology: Trust, Suspicion, and the F-Scale
      (pp. 81-84)

      As a central part of their monumental studyThe Authoritarian Personalityin 1950, T. W. Adorno and his co-workers developed one of the prototypes of the personality inventories now so familiar to us all. The F-scale inventory was a series of statements to which subjects were to respond by writing a number from 1 (strongly disagree) to 7 (strongly agree). The statements were designed to test personality variables which, the authors argued, underlay susceptibility to authoritarian ideologies. Here are the personality variables included, together with corresponding sample items from the inventory:

      Conventionalism:“One should avoid doing things in public which...

    • 14. Strategic Moves
      (pp. 85-92)

      In our analysis of non-zero-sum games up to this point, we have required that the players choose their strategies simultaneously and not communicate with each other beforehand. Of course, games in real life may not be like this. In this chapter we will consider some of the things which might happen when one player can move first and make his move known to the other player, or when the players can talk to each other before they move. Commitments, threats, and promises become possible. The classic analysis of these kinds of “strategic moves” is [Sendling, 1960], which I strongly recommend....

    • 15. Application to Biology: Evolutionarily Stable Strategies
      (pp. 93-101)

      The idea of an evolutionarily stable strategy (ESS), first introduced by John Maynard Smith and G. R. Price [1973], is a powerful explanatory idea in evolutionary biology. It is especially applicable to the study of behavior, and has found an important place in modern sociobiology. The basic idea is this. Because individual members of a biological species have similar needs, and resources are limited, conflict situations will often arise. In these conflict situations, there are many different behavior patterns (strategies) which individuals might follow. Which ones will they choose? Since the question involves behavior in conflict situations, a game-theoretic formulation...

    • 16. The Nash Arbitration Scheme and Cooperative Solutions
      (pp. 102-111)

      In our analysis of non-zero-sum games up to this point, the players have played non-cooperatively. Each has tried to do the best possible for himself by choosing strategies or by making strategic commitments, threats or promises. In this chapter we will consider a different approach. Imagine the players sitting down together to decide what is a reasonable or fair outcome to the game, and then agreeing to implement that outcome. Alternatively, imagine that the players call in an impartial outside arbitrator to determine a reasonable and fair outcome, and agree to abide by her decision. What principles should guide the...

    • 17. Application to Business: Management-Labor Arbitration
      (pp. 112-117)

      The management of a factory is negotiating a new contract with the union representing its workers. The union has demanded new benefits for its members: a one dollar per hour across-the-board raise, and increased pension benefits. In turn, management has demanded concessions from the union. Management would like to eliminate the 10: 00 a. m. coffee break, which has proven to be excessively costly as workers straggle slowly back to the assembly line, and to automate one of the assembly line checkpoints. The union opposes both demands, especially the automation, which would eliminate union jobs. The dispute has not been...

    • 18. Application to Economics: The Duopoly Problem
      (pp. 118-124)

      Aduopolyis a situation in which two companies control the market for a certain commodity. The duopoly problem is to decide how the companies in a duopoly situation should adjust their production to maximize their profits. In this chapter we will use a simple example to compare four different “solutions” to the duopoly problem. Some of the solutions involve calculus—at least knowing how to differentiate polynomials and the fact that a maximum value of a function occurs at a point where its derivative is zero. If you know calculus, you can calculate these solutions along with me; if...

  6. Part III N-Person Games

    • 19. An Introduction to N-Person Games
      (pp. 127-133)

      Until now, we have dealt only with games played between two players. In our modern interconnected world, such games are rare. Most important economic, social, and political games involve more than two players. We will now turn our attention ton-person games, wherenis assumed to be at least three. We will find that with three or more players, new and interesting difficulties appear.

      To begin our analysis, let us consider the simplest possible case, a three-person 2 X 2 X 2 zero-sum game. Game 19.1 is an example.

      The three players are Rose, Colin and Larry (Larry chooses...

    • 20. Application to Politics: Strategic Voting
      (pp. 134-138)

      In the 1980 United States presidential election, there were three candidates: Democrat Jimmy Carter, Republican Ronald Reagan, and Independent John Anderson. In the summer before the election, polls indicated that Anderson was the first choice of 20% of the voters, with about 35% favoring Carter and 45% favoring Reagan. Since Reagan was perceived as much more conservative than Anderson, who in turn was more conservative than Carter, let us make the simplifying assumption that Reagan and Carter voters had Anderson as their second choice, and Anderson voters had Carter as their second choice. We then have the situation

      If all...

    • 21. N-Person Prisoner’s Dilemma
      (pp. 139-144)

      In Chapter 191 mentioned that not every n-person non-constant-sum game can be analyzed in a satisfactory way using the characteristic function form. In this chapter we will look at a particularly important type of a game which cannot. Game 21.1 is a three person example.

      This game is symmetric for the three players, and strategy D dominates strategy C for all of them. The unique equilibrium is DDD with payoffs (—1, —1, —1). This is a Pareto inferior outcome, since CCC with payoffs (1, 1, 1) would be better for all three players. We recognize the game as a...

    • 22. Application to Athletics: Prisoner’s Dilemma and the Football Draft
      (pp. 145-149)

      In ann-person Prisoner’s Dilemma game, if all players rationally pursue their own best interests, all players end up worse off than if they had all followed some individually less rational line of play. We have seen that this type of situation appears often in the course of human interactions. One surprising place it can appear is in sequential choice procedures. In this chapter we will analyze one well-known sequential choice procedure, the professional football draft in the United States.

      In football, basketball and other professional sports, teams choose new players by a draft system which involves sequential choices. From...

    • 23. Imputations, Domination, and Stable Sets
      (pp. 150-160)

      In this chapter we begin the search for solutions ton-person games in characteristic function form. I should tell you at the outset that the situation is going to be at least as murky as it is for two-person non-zero-sum games. Forn-person games there are a number of different useful and illuminating ideas of what a solution might be, but none of them is completely satisfactory in all situations.

      Suppose we have ann-person game in characteristic function form (N, v). We will assume that the game is superadditive. The two questions we would like to answer about such...

    • 24. Application to Anthropology: Pathan Organization
      (pp. 161-164)

      One of the earliest applications of cooperative game theory in anthropology was Fredrik Barth’s study of political incentives among the Yusufzai Pathans of northern Pakistan. Barth used a simple game-theoretic model to clarify the logical structure of these incentives and explain behaviors which at first seemed puzzling. I’ll describe the situation as it was when he studied it in the 1950’s.

      The Yusufzai Pathans (pronounced “Pun-táns”) occupied the agriculturally rich lowlands in and around the Lower Swat Valley, near the Khyber Pass in the Northwest Frontier Province of Pakistan. Almost all land was owned by a dominant male aristocracy, and...

    • 25. The Core
      (pp. 165-170)

      Von Neumann and Morgenstern’s idea of a stable set (Chapter 23) was historically the first proposed solution for games in characteristic function form. It was introduced because in essential constant-sum games, no single imputation is stable: every imputation is dominated by some other imputation. However, in non-constant-sum games there may be undominated imputations, and in the early 1950’s Gillies and Shapley pointed out that the set of all undominated imputations in a game is an object worthy of study. Gillies called it thecoreof the game.

      Definition. Thecoreof a game in characteristic function form is the set...

    • 26. The Shapley Value
      (pp. 171-176)

      The two solution concepts forn-person games in characteristic function form which we have considered so far, von Neumann-Morgenstern stable sets and the core, are based on possible divisions ofv(N)which might result from coalitional bargaining. In particular, they are based on the concept of domination. The core is the set of undominated imputations. A stable set is a more complicated set of imputations meeting more subtle conditions of internal and external stability. In both cases the solution is a set of imputations rather than a single imputation, since coalitional bargaining is too complex for us to expect to...

    • 27. Application to Politics: The Shapley-Shubik Power Index
      (pp. 172-184)

      In a voting body, the voting rule specifies which subsets of players are large enough to pass bills, and which are not. Those subsets which can pass bills are calledwinning coalitions,while those which cannot are calledlosing coalitions.We can model a voting body as a characteristic function form game by assigning a value of 1 to all winning coalitions and 0 to all losing coalitions. The resulting game, in which all coalitions have a value of either 0 or 1, is called a simple game. A simple game is completely specified once we know its winning coalitions,...

    • 28. Application to Politics: The Banzhaf Index and the Canadian Constitution
      (pp. 185-189)

      When John Banzhaf wished to demonstrate the inequity of the Nassau County Board (Chapter 27, Exercise 7), he did not use the Shapley-Shubik index, but devised his own power index. Banzhaf reasoned that a voter only has a direct effect on the voting outcome when he is a swing voter in some winning coalition, and hence a voter’s power should be proportional to the number of coalitions in which that voter is a swing voter. The resulting index is known as theBanzhaf index,and it is sometimes used as an alternative to the Shapley-Shubik index.

      To see how the...

    • 29. Bargaining Sets
      (pp. 190-195)

      If we plot the core conditions as in Figure 29.1, we see that the core is empty. This game is not constant-sum, so not equivalent to Divide-the-Dollar, and its von Neumann-Morgenstern stable sets are complicated. The Shapley value is easy to compute (Exercise 1), but is not meant to describe an outcome which might actually occur if this game were played. Furthermore, the solution theories for non-constant-sum games which we have considered so far assume that the outcome of this game will be an imputation, i.e. that the players will divide the full 105 units. However, it is possible to...

    • 30. Application to Politics: Parliamentary Coalitions
      (pp. 196-201)

      In a parliamentary democracy with more than two major parties, it is common that no single party will have a majority of seats in the parliament. Hence a majority government must be formed by a coalition of parties. In this chapter we will consider several game theoretic approaches to the study of parliamentary coalitions.

      As an example, consider the 1965 parliamentary election in Norway, which gave results

      It takes 75 members to form a coalition government. Can we predict which parties formed the government?

      In the weighted voting game

      there are five coalitions which areminimal winning,in the sense...

    • 31. The Nucleolus and the Gately Point
      (pp. 202-208)

      The Shapley value of ann-person game in characteristic function form is a single imputation which might be proposed as a “solution” to the game. It is based on axioms embodying a concept of fairness, rather than on bargaining considerations. In this chapter we will look at two single-imputation solutions based on concepts of bargaining.

      The first, proposed by David Schmeidler [1970], is known as thenucleolus.Recall that the core consists of all imputations x = (x1,... , xn) which satisfy

      $\sum\limits_{i\varepsilon S} {{x_i} \ge } v\left( S \right)$for every$S \subseteq N$.

      If the core is empty, no imputation satisfies all of these constraints. However, we...

    • 32. Application to Economics: Cost Allocation in India
      (pp. 209-212)

      In this chapter we will consider a problem of cooperative hydroelectric power development in southern India in the 1970’s. The analysis is due to Dermot Gately [1974a]. The players were the four states comprising the Southern Electricity Region of India: Tamil Nadu, Andhra Pradesh, Kerala, and Mysore. To simplify calculations and because of economic and geographical similarities, Gately considered Kerala and Mysore as a single state. He then used an integer programming model to predict costs of expanding and operating the electric power system in the region under five scenarios: each state acting individually, three possible pairs of two states...

    • 33. The Value of Game Theory
      (pp. 213-216)

      When I ask my students who enjoy mathematicswhythey enjoy it, one of the most common answers I get is that mathematics “gives exact answers.” It offers comforting certainty as opposed to the tenuousness and ambiguity of the humanities and the qualitative social sciences. It is, then, a natural hope that when mathematical reasoning is applied to the study of human interactions, as it is in game theory, exact answers will emerge. What was cloudy and ambiguous before will become clear and certain. Hope like this animates A. H. Copeland’s famous review of von Neumann and Morgenstern’sTheory of...

  7. Bibliography
    (pp. 217-224)
  8. Answers to Exercises
    (pp. 225-240)
  9. Index
    (pp. 241-244)
  10. Back Matter
    (pp. 245-245)