CONnICT AND CO-OPE %ON

Structure fage No. 12.1 Introduction 34 Objectives 12.2 Some Games and its Characteristics 12.3 Two-Person, Zero-Sum Game 12.4 Co-operative and Non-do-operative "Prisoner's Dilemma" Game "Battle of Sexes" Game Duopoly 12.5 Somc Generalizations 12.6 Some Applications and Limitations 12.7 Summary 12.8 Solutions/Answers

12.1 INTRODUCTION

Life is full of conflict, coinpetition and co-operation. Numerous examples involving adversaries in conflict include political campaigns, advertising and marketing campaigns by competing business firms, military battles, parlour games etc. A basic feature in many of these situations is that the final depends primarily upon the combination of strateEies selected by adversaries. Game Theory is a mathematical theory that deals with the general features of competitive situations like those in a formal abstract way. Game theory was developed in 1940s to analyze competitive situations in business, warfare, and social situations. It deals with how to make decisions, when in competition with an aggressive opponent. There are situations in which both competition and co-operation figure, For example, let us consider the situation in which a trade union and employer negotiate wage rates, productivity etc. of an industry. Both the trade union and the employer have some objectives. They negotiate, aiming to maximize the achievement of these objectives. In order to attain these desired objectives, the two sides may be in direct competition, for example over wage rates, whereas to achieve others, it may be in both's interest to co-operate, for example to improve the viability of the company. In this unit, we want to use modelling to analyse quantitatively situations in which competition and/or co-operation occur. The techniques of game theory can be applied to any system involving interdependence. This not only includes parlour games but also other game-like systems which occur in economics, politics, psychology, sociology etc. Also game theory analyses quantitatively some systems in which interdependence and hence conflict and co-operation are present, to find out the extent of co-operation and extent of conflict. Game theory is divided into two branches, co-operative and non-. The distinction can be fuzzy :at time. In non-cooperative game theory the unit of analysis are the individual participants in the game who are concerned with themselves, without any compassion for the other player, subject to clearly defined rules and possibilities (the set and pay-off matrix discussed later), If individual happens to undertake behaviour, that in common parlance would be labelled as "co-oper;tion", then this is done because such co-operative behaviour is in the best interest of each individual singly. Each fears retaliation from others, if co-operation breaks down. Whereas, in co-operative game theory, most often, the unit of analysis is the group or the coalition which will be elaborated at a later stage. When a game is specified; part of the specification is what each group or coalition can achieve without reference Lo how the coalition would effect a particular outcome or result. -file resull ol~allo\\!irl;; ~rornn~l~nicatiolldepcnd on thc characters of the individual, Conflict md Co-operation malung it impossible for any general theory to provide meaningful solutions as you will see later on. Hence, garrrc theory has tended to by-pass this problcrn by studying co- operative games in which absolute binding agreements on the strategies used by the players, arc cntercd illto bcfore the play begins. r;irst, we discuss hasic clcments of tl~cgame theory. Next we concentrate on the basic model for two-pcrson zcro-sun1 gamcs in some dctail, illuslrating non-cooperative games. Here, we discuss some r~~cthodsof solvi~~gsuch gamcs. Then we briefly indicate the nature of co-opcrative game theory ilnd non-zero sum game by discussing celebrated garrics like "'Prisoners' Dilemma", "Battlc of Sexes" and Duopoly. We illustrate real world situations, which can be rnodclled by thcse games. Further we conclude by discussing somc generalizations of thc game theory and some of its limitations or drawbacks.

Objectives

After reading this unit, you should be able to lcnrn: s about stratcgics of difi'crcnt players, thc pay-ofl' matrix or table of pay-off of several gamcs; how to solve two-person zero-sum gamcs by various methods; about Prisoncbr's Dilcrnma gamc, Battlc of Sexes game and Duopoly game ant1 thcir: cliarac~cristics; s about the gencral sort of real world situation which rccurs in many contcxts of econoniics/social condition, which can bc modcllcd by P~.isoncr's Dilcrnma game;

Q about ~hcgcncral sort oT real world situation, which can be rnodclled by tlie Battle of Sexcs gamc;

QI about dil'l'crent directions in which gnmc tlicory has been generalized; about :iomc areas of' applications and the limitations of' this theory, as far as modelling is concerned.

11.2 SOME GAMES AND ITS CIIARACTEMSTICS

We now consictcr somc cxamplcs and discuss sonlc spccific games. In the process, wc introducc thc rclcvant tcr~ris,and notations nrld biisic charactcristics of the gamc theory. Example 3 : (Scissors, Paper, Stonc Gzlmc). This is a tlqaditionnl cliilclrcn's game for two playcrs - player A and player B. Each player choosrs one of the thrcc objccts, scissors, papcr or stone, and the players simultancously clcclarc thcir cl~oicc.'l'hcre arc thrcc options and thesc arc called the strategies or choiccs availnblc to cach, callcd: stone, papcr, and scissors. Dcpcnding on what each child select, thc game is cithcr won by one child or the othcr, or it is a draw. If both choose thc same option, thc outcome is a draw. If one chooses stone and the other papcr, the sccond wins as papcr covcrs stone. If onc chooses slonc and thc second scissors, thc first wins sincc stone blunts scissors. IS onc chooses paper and tlie second scissors, the sccond wins as scissors cut papcr. Suppose winncr gains one point (pay-oSF = I), tlic loser loscs onc point (pay-oSf = -1) and no pvi~~tsarc awarded (pay-of = 0) iT a draw occurs. Obviously, the resulting points gaincd by the playcrs are il function of the stratcgics adopted by cach of thc playcrs. The result or the outco~necan bc exhibited in a matrix lor~nor in a table, known as pay-off matrix or table or outcome maxtrix. A playcr's choice is called a strategy. For this game, the pay-off matrix is prescntcd bclow: Player B Strategy Scissors Paper Stonc Scissors 1 Player A Paper 0 -1 Stone L 1 -1 0J1 Socio-Economic Environment We denote playcr A's pay-off matrix by UA, given above and the entries represent the points gaincd in the game by A, We list child A's strategies as rows in the matrix and child B's strategies as columns. Child B's pay-off matrix is - UA, i.e. U, = -U,. You will see later that this is a two-person zero-sum game. Before discussing some more games, we discuss the characteristics of the game. A game, in game theory is characterised as follows: 1) Each player has a well-defined set of possible courses of action. These are called pure strategies. 2) The outcome of the game, or the probabilities of the outcomes are conipletefy determined by a choice of strategies by each player. 3) Each player has an order of prefetence among the possible outcomes of the game. 4) Every player has complete knowledge of the strategies, their outcomes and preference of all the players. A strategy is a predetennilled rule, that specifies completely how one (player) intends to respond to each possible circumstance at each stage of the game. For exan~ple,a strategy for one side in a play, would indicatc how to make the next move for every possible position on thc board. Obviously, the total number of possible strategies (moves) would be astronomical. Applications of game theory gcnerally involve far less complicated competitive situations than that encountered in a chcss play. The outcome matrix or pay-off table shows gain (positive or negative) for player A that would result from each combination of strategies fot the two players. Generally it is given only for player A in zero-sum game, because the table for player B is just tl~e ncgative of this one, For non-zero sum game the outcolnc matrix for hoth players are given. Very often, the numbers (or pay-offs) in a pay-off matrix do not represent money (profits or costs for. example), but utility. A utility is a number, tllal measures the satisfaction (or lack of it) that results from a certain action. The number must be assigned by each player, depending on how he or she feels about the situatiofi. In the game "Scissors, Paper, Stonc" we have such a pay-off matrix.

Example 2: (Duopoly) Two firms A and B, assumed identical for simplicity, producc the same commodity. Each firm is able to produce only 0, 2, 4, ot 6 units because of their production methods. The cost of producing qA units is Rs [4 + (qi - 9cli + 27qA)/2]. Firm B has identical cost function, i.e. Rs [4 + (d-gq; -i- 27q,)/2] for producing qB units. The revenue for firm A is Rs pq,, where p is the price of a unit. Suppose the demand function is given by D (p) = (72 - p)/4 The total production of the two finns is qA + qB. WC assullle that demand equals supply (~narketis in equilibrium), i.e.

~IA-i- 9~ = D (p). The strategies for A and B are to produce 0, 2, 4 or 6 units. Thc outcolnes are the profits of the firms. Below we have the outcome rnatrix U of A, in Rs per period. (calculation for one element of the pay-off matrix is illustrated). B's strategies 0 2 4 G

A's strategies U = 2 1 111 95 79 63 1

6 257 209 161 113 ~

You may be wondering how we got these entries or how the pay-off matrix is written? It is easy. We illustrate here, the calculations of one element of Llw pay-off matrix. ..

- Let q, = 2 and qg = 4 Conflict and CO-operation q, + q, = 2 + 4 = D(p) = (72 -~)/4.

The profit for A is pqA - I4 + (qj- 9~12+ 27q,,/2)

This gives the uz3entry in the pay-off matrix U of A. Since A and B are identical, tho game is symmetrical. (A game is said to bo symtnetric for two players A and B if their strategies are the same and a swap of the strategies results in a swap in the outcomes). B's outcome matrix is IJ~,the transpose of U. Is it a zero sum game?

Fxample 3: (Prisoner's Dile~n~naGame) The police. have apprehended two individunls whom they strongly suspect of a crime (ppd who in faal committed Uie crime together). But the police lacks the evidence necessary to convict them and must release the two prisoners unless, at least, one of them confesses. They put the men into separate cellslrooms for interrogation and separqtely makc the same proposition to each of them. IfA confesses and B does not confess then A will get off with only a warning. But if A does not confess and I3 does, then A will get at least 8 years imprisonment. It is in onp's own interest to confess. If both A and B rcfuses to confess, the11 hoth will go to jail/prison for 2 years. If A and B both confess they will get 5 years imprisonmcnt. The prisoner'. are both aware of these penalties. The strategy for each prisoner is obvious. Either he conl'esses (C) or refuses (R) to cqnfess. The pay-off matrix is as follows: Prisoner B's Strate~ies C R Prisoner r-5(-5) ,(-,) 1 A's strategies I I

Example 4: (Battle of Sexes Game) I This game is about a husband aid wife, who are trying to decide how to spend an evening. The man would likc to attcnd n football ~natch(flood -lit), the woman, a cinema. But each would r~therattend an entertuinmenl with tho other than not, and this is more important to each, dhan attending his or her preferred fonn of entertainment. The pay-off matrix is as follows: Husband Go to football Go to cinema Got to wife O(0) 1

Cinema 5(4) It is not a zero-sum game. This game is representative of many situations in which, two (or more) parties seek to co-ordinate their actions, although they have conflicting preferonczs, concerning which way to co-ordinate. We give an exa~pleto illustrate how it helps in, decision making, say in investment. Socio-Economic Environment Example 5: An investor has Rs. 20,000 to invest in stocks. She has two possiblc strategies: buy conservative blue chip stocks or buy highly speculative stocl

Buy Bluc-chip stocks : 25,000 x 0.7 -1- 18.000 x 0.3 = 22,900 Buy sl~eculativestocks : 30,000 X 0.7 + 11,000 x 0.3 = 24,300 Here the best strategy is to buy speculative stocks. d) The expected returns are Buy Blue-chips : 25,000 x 0.2 + 18,000 x 0.8 = 19,4,00 Buy speculalive ; 30,000 x 0.2 + 11,000 x 0.8 = 14,300 best strategy : buy blue chip. In the next section, we concentrate on a particular type ol' galnc called "Two person zero-sum game" and also discuss some of the methods of solving these ganics.

I 12.3 TWO-PERSON, ZERO-SUM GAME ---

The game which involves thr ?NO players A and 13, is called a two-l)erson game. Paper, Stone and Scissor game, given in Example 1, is an exalnple oS two-pc!rscln gamc. There you must have noticed that no money or utility number enters the ganc frotn the I I outside. Whenever one player wins, the other player loses. Such :I g:lme is callecl a zero-snm game. The Duopoly game discussed in Exatnple 2, is not a zero-sum ganlc. The stock inarket and Investor is not a zero-sum game. Stocks can go up or down 1 according to outside forces. l'herefore, it is possible that all investors can makc or Iosc money. I I 38 We sllall now discuss two-person zero-sum games in detail. Each player can 11;lve many I 1 dii'lcrenl oplions. In an In X n matrix galllc (i.e. with 111 X 11 pay-OFmatrix), the 1]laycr A Co~itlictand Co-operntion lias m str;ltegies (rows) and player B has n slratcgics (columns). A ~vi~iiaryobjcctivc of non-cooperative gamc theory is lie develol~mentof rational crilcria for selecting a strategy. Thc l'ollowing two key assumptions arc made: 1) HoLh players are ralionnl

2) Both playcrs choose thcir slralcgies, solcly to promotc thcir own welfare, with no compassion for tl~copponcnl.

To formulate any problem :IS n Lwo-pcrson, zero-sum gnmc, we must idcnlify the two players, the stratcgics for each player, ay-ofruldthc o~~tco~iic/pay-ofry-ofmatrix.

Consider he follow~ngcxample of 3 X 3 matrix gamc. The best possible strategy lor each plnycr is to he dctermincd. Thc game is dcl'incd l>y the following 3 x 3 pay-off lnalrix

Player A 2 0 3, --" 8 11: ~cro-sumg;niic, the pay-ol'ls to tl~cplnycr A i\rc shown in Lllc pay-off matrix, i.c. U, is glvcn. Tlic pny-oil' nlatrix I'or [hc plnycl. B is - U,, (U, -I- U,, = 0). From the playcr B's viewlloinl, slrotegy 2 is I~cltcrtlinn sll-nlcpy I, no mnllcr which str;iLcgy ~I;IYCI. A sclcc\s. Tl~scan bc seen by coliiparing (lie I'irst two columns of tlie ~ny-off111all.i~ UN or - LI,. TI' A chooscs row I,. rcccivi~~gICs. G Srom A is bcllcr than rcccivi~igRs. 3. In row 2, breaking cvcn is hcttcr than paying Rs. 3 and in row 3, geltir~gRs. 4 horn A is bcltcr than paying Rs. 5. Tl~crcl'ore,B should ncvcr scloct stratcgy 1. Strategy 2 is said to doniinatc str:\legy I iind clominatccl stralcgy can IIC r~~novcclfro111 thc pay-ol'f matrix, resulting in tlic I'ollowing reduccd motrix.

111 [his 3 x 2 matrix nbovc, ncithcr pl:lyer now has a clon~in;~tcdstratcgy. A row I'or A cloniinalcs nnollicr row il' evcsy cnlry in the first laow is cqu:~l or largcr Llian llic corl-esponding entry in tlic sccond I-ow. For a colu~nnfor B to dominate anolher column, each entry must hc smnllcr than tlic corresponding entry in tlie othc17 colurnn (ol' the pay-off malrix UA). More P~CCISCI~,(i) If all lie clc~ncntsof n raw, say, ktli, arc ICFS thiln or cq~~alLC) tlic corresponding clements oS any olhcr row, say rth, i.c., uk, 5 uli for all i and ilk, c u,, for :IL least onc i, then kth row is dominatccl by lhc rlh row. (ii) IS all the clc~ncntsCII' a column, say kth, arc grcnter than or cqual to tlic corresponding elements of any othcr col~~~nn,say rth, i.e. LI,~2 uir for all i i~nduIk >ui, Sor at least one i, the ktli. colomn is dominated by tlie 1111 column. In such cases, one can rcducc tlic size of the pay-off m:llrix by deleting those stmtcgies which are dominated by tlic orhcrs. l'hc value ol' Ilic game and the non-zcro choice of probal~iliticsrcmain uncli;~ngcdcven aftcr t11c dclction oS such rows and columns. Either playcr may havc dominated stmtcgics, In fact, after a dominated strategy lor one playcr is removecl, lhc other playcr may, thcn, liavc a dominated stratcgy wIiel*c there wiis none bclbrc. The conccpt of a dominatcd st~te.gyis a very uscSirl one for reducing the size of the pay-ol'f matrix. 'Thc dominance properly is not always hascd on thc superiority' of pure stmtegics only. Sncio-Econoniic Environment A given strategy can also be said to be dominated, if it is infkrior to an average of two or Inore other pure strategies. More specifically, if some convex linear combination I of some qther rows dominates the ith row, then the it11 row can be deleted (hulj + (1-h)umj 2 uij etc.). If ith row dominates the convex linear combination of some other ~ rows, one of the rows involved in the convex combination may be deleted. Similar arguments hold for columns. This is called tt,e modified dominance property. I Strictly Determined Game The objective of game theory is to find optimum strategies. The strategies that are most profitable to the respective player (say ith row and jth column) are called optimum strategies. The pay-off to player A that results from each player's choice of the optimum strategy is called the value of the game (aij of the pay-off matrix UA). The position of any such entry is called a saddle point. The simplest strategy for a player is to consistently choose a certain row or column. Such a strategy is called a pure strategy. In a game with a saddle point, the optimu~n pure strategy for player A is to choose Lhe row containing the saddle point, while the optimum pure strategy for B is to choose the column containing the saddle point. The optimum pure strategy in this game, for A (the row player), is found by identifying the smallest number in each row of the pay-off matrix and the row having the largest sucli number gives the optimum strategy. That is, we find Fax win aij where 11 U, = [aij]. This is called the lower value of the game (y). For the player R, the optimum pure strategy is to identify the largest number in each column of the pay-off matrix, and then choose the column producing the smallest such number, that is, we find min mpx aij. This is the upper value of the game (9.ll~is so -called J1 criterion is a standardcriterion proposed by game theory for selecting a strategy. It states that, each player should play in such a way as to minimize histher maximum losses, whenever the resulting choice of strategy can not be exploited by the opponent to then improve histher position. If

rnax gin aij= mjn max aij = v (y = i = v discussed later), then the position 11 11 of any such entj is called a saddle point. In general, the pay-off to player A, when both players play optimally is referred to as the value of the game. A game with a saddle point is called a strictly determined game. A game that has a value zero is said to be a fair game. We illustrate this by the following example.

Example 6: Find the saddle point in the following game.

Strategy 1 A 2 3 4

Solution: For the pay-off matrix maximum value and minimax value are shown below.

' B Strategy 1 2 Minimum

Maximum 3 7 t Minimax Notice the interesting fact that the sane entry, 3, in this payoff matrixltable yields both Conflict and Co-opcration the maximin and minimax values. The reason is that this entry 3 is both the minimax in its row and maximum in its column. The value of the game is 3. Optimal strategies are (4, I), that is, A's optimal strategy is 4th row (strategy 4) and B's optimal strategy is 1st column (strategy 1). (4, 1) position is the saddle point. The following example shows that not every game has a saddle point.

Example 7: Consider the game having the following pay-off table.

Strategy 1 2 3 4 r-2 4. - l 6l

Check whether a saddle point exists or not.

Solution: By dominance method, the reduced pay-off matrix is Minimum

A [-: :]1: + Maximin Maximum 1?. 4 Minimax

Notice that maxiniin value (-1) and the millrnax value (3) do not coincide in this case. The result is that there is no saddle point. The impasse presented by such a game can be resolved by allowing the players to sclcct their strategies on a probabilistic basis. Let us see how this can be done matl~ematically.

Mixed Strategies Wllenever a game does not possess a saddle point, game theory advises each player to assign a probability distribution over his or her set of slrategies. That is, it will be necessary for both players to mix their strategies according to some previously determined probabilities,, To express matheinntically, let pi = probability that player A will use strategy i (i = 1, 2, ..., m) qi = probability that player B will use strategy j (j = '1, ..., n) where m and n are the respeclive numbers of available strategies. What each player needs is a rule which determines how often a pure strategy should be chosen but does not determine the actual strategy at each play. He ncds to know the probability that a particular strategy should be chosen and the actual cl~oiccis then left to a lottery device giving the probability. For example, with three rows with p, = 0.3, p2 = 0.1, and p7 = 0.6, he may assign the numbers 0 through 2 to the first strategy, 3 to the second and-4 through 9 to the third. One digit number is selected by a random process. This would result in the two strategies being used about equally over the long run. In case of three strategies, one could select strategies if 1, 2, and 3 comes up by rolling a die. Let the pay-off matrix for 2 x 2 game be

Player B

91 q2 Player A

Maximum P2

I I Socto-Economic Environment Let the player A choose row 1 (strategy 1) with probability p, and row 2 with probability p2, such that p, + pa = 1. Let player B choose column 1 with probability ql and column 2 with probability q2 such that q, + q2 = 1.

The probability of choosing row 1 and column 2 is P (law 1, colutnn 2) = l? (ROW ,I) P (column 2) =: pIq2, due to indepdenct. The probabilities of each possible outcome are given in tabular form below:

Outcome hobabilities Pky-off for A of 0u.tcome

Row 1, Column 3 Row 1, Column 2 Row 2, Column 1 Row 2, Column 2

I The expected value for this game is

Expected value = [pl, y2] K::ll 1L:g

In case of m x n payoff mabix [aij]

Although no completely satisfactory measure of performance is available for evaluating mixed strategies, a very useful one is the' expected pay-off. Applying the probability theory definitions of expected value (done nbove), this quantity is rewritten as m n Expected pay-off = $, piql (1) I=l j=l qj is the pay-off if player A uses pure strategy i and player B uses pure strategy j. Using this measure, gbetheory extends the concept of the minimax criterion to games that lack a saddle point and thus need mixed strategies. In this context, the minimax criterion says that a given player should select the mixed strategy that minimizes the maximum expected loss to himself or herself. Equivalently, when focussing on pay-offs (player A) rather than losses (player B), this criterion says to maximin instead, i.e. maximize the minimum expected pay-off to the player. By the minimum expected pay-off, we mean the smallest possible expected pay-off the opponent can result from any mixed strategy with which the opponent can counter. Thus, the mixed strategy for player A that is optimal according to this criterion, is the one that provides the guarantee (minimum expected payoff) that is best (maximal). The value of this best guarantee is the maximum value, denoted by Y. Similarly, the optimal strategy for player B is the one that provides the best guarantee, where best means minimal, and guarantee refers to the maximum expected loss that can be administered by any of the yponent's mixed strategieb. This best guarantee is 'the minimax value, denoted by V, When only pure strategies were used, games not haying a saddle pint turned out to be unstable (no stable solution), 'rho reason was essentially that Y < V, so that the players would want to change their strategies to improve their positions. Similarly, for games with mixed strategies, it is necessary that y = for the optimal solution to be stable. Fortunately, according to the minimax theorem of game theory, this condition always holds for games. Minninaax Theorern: If mixed straregles are all(~\~ed,the pair or mixetl strategies that is optl~~~alaccording to the minimax c~iterionprovldes a strble solution with = V = V (the value of thc gdrne), so that neitha playel- can do better by unilaterally changing his strategy.

~'iluwwe show Yluw to find tlrz optinla1 mixed stlategy for each player. 'ICwci method:s \vi]l be discussed. First one is a graphical procedure that may be used wlr(!never of ghe gliyiyess BRMS only two (nntlo~ni~antcd)parc strategies. When larger games are involved, ttae usual method is to wansfonn the problem into a linear programming problellra that would then he sollved by the sitrlplcx raiathod. Here we will not go into details of the sirnplex method. In some simple cases, solution of the LPB can bc obtained by graphical method. a) Graphisall Solution Procedure and Linear Programming Method Let us consider any game with mixed strategies, such that, after eliminating dominated strategies, one of the players has only two pure strategies. To be specific, let this player be A. We illustrate this procedure by an excunple given in this section later. (Example 8). Let us now discuss the linear programming method. In such a game, the decision problem of each player is to selecl an optimal, set of probabilities. A fears that B will discover his stratcgy and that B will select a strategy of lris own that will maximize his expected outcome, that is, minimize the expected outcome for A. B has similar fears about A. l'he probabilities which the players employ a.re defined as optirnal if

and 2 aij qj < V i = 1, ..., m j=l where V is defined as the value of the gatne. The relations (2) state that A's expected profit is at least as great as V if B einploys any of his pure strategies with a probability of one, and then relations (3) state that B's expected loss is at least as small as V if A employs any of his pure strategies with a probability one. A fundamental theorem of game theory states that n solution i.e. value of ps and qs that satisfy (2) and (3), always exists, and that V is unique.

The optimal strategies for the players and the value of the game can be determined by converting the game problem to a linear programming prclblem.

Let us consider the case V > 0. Let us define the variables xi = -q' , i = 1, .,., n, for n V the player B. By qi = I , we have i =l 1 -= XI + '.' + X, V

B desires to make his maxirnl~mexpected loss as small as possible, or, equivalently he desires to make -1 as large as possible. v

Equivalent linear programming problem for B 'is

1 Max -v = x, + x2 + ... + x,

Subject to a,lxl + a,,x2 .. . + ainx, 5 1, i = 1, ... m.

xj 2 0 ('j = 1, .., ,n) n Socio-Economic Environment iobtained from aij q, L V i = I, . . .. m j =1 qj by dividing by V and using xj = -v I.

e Similarly for A, let us define the variables

1 Again we have -:= y , + y2 + . . . t y,,. Y '

A desires to make his minimum expected profit as large as possible or equivalently, he 1 desires to make -v as sr~lallas possible. A's LPP equivalent Ts the following:

Mjh - = y, + ... + Ym V Subject to aljyl $- azj y2 +...+a,,,jyYm 2 1 j = 1, ..., n. This is the dual of the LPP for B given above.

When larger games are involved, the.usua1 method is to transform the problem into a linear programming problem which can then be solved by the simplex ' method. It is important to remember that the use of mixed strategies is meaningful only if a game is repeated many, many times. In the next example, we illustrate the graphical solution method mentioned in the beginning.

Example 8: Solve the game having pay-off matrix Min.

7 L 3 + Maximum value Max. 7 8 Minimum value J Minimax value I

Solution: This game does not posses a saddle point. We solve this by graphical method and by equivalent LPP method. Player A needs to develop an optimum strategy - a strategy that will produce the best possible pay-off no matter what B does. Suppose A chooses strategies I and 2 with probabilities p and 1 - p. If player B chooses Column 1, then A's expectation is E,(p) = 2. p + 7 (1 - p) = 7 - 5p If player B chooses Column 2, then A's expectation is E2(p) = 8. p + 3 (1 - p) = 3 + 5p.

Fig. 1 NOW we plot these expected pay-off lines El and 5 on a graph as shown below. For Conflict and Co-aperation any given value of p. and (q,, q2), the expected pay-off is given by. Expected pay-off = q, (7 - 5p) + q2 (3 + 5p) ~hus,given p, the minimum expected pay-off is given by the corresponding point on the "bottom" line. According to the minimax (or maximin) criterion, player A should select the value of p giving the largest minimum expected payoff, so that ' y = V = max {min (7 - 5p, 3 + 5p)j

Therefore, the optimal value of p is the one at the intersection of the two lines 7 - p and 2 3 + 5p. Solving i.e., when 7 - 5p = 3 + 5p, we have p = - . 5 A needs to maximize the smallest amounts that can be won by choosing the point of intersection itself (point C), the peak of the heavily shaded line in the figure above. A's optimal probabilities are p = 0.4, 1 - p = 0.6,

How should B play? Suppose B chooses strategies I and 2 wilh probabilities q and 1 - q respectively. Then El(q) = 2.q + 8 (1 -q) E,(q) = 7q + 3 (1- q) Minimax occurs when the two lines meet, that is; when 8 = 6q = 49 + 3 => q = 0.5. Therefore, ql = 0.5, q2 = 0.5 and V = 4 x 0.5 + 3 = 5. One can draw a Figure likc above corresponding to this Case. Now, we solve this game by solving equivalent LPPs for the two playcrs. The LPP equivalent (for A)

Min -1v = Yt+Y,

Subject to Basic Feasible set is bounded from below

~YI+ ~YZ 2 1 8yl + 3y2 2 I T

= .5 + 0 = 0.5 at Q Hence Min (y, + y2) = 0.2

PI = VYI = 5 x .08 = 0.4 p2= Vy2 5 x -12 = 0.6 For B, the equivalent LPP is - Max x, -tx, Subject to: 2x, + 8x, 5 1 7x1 + 3x2 5 1 Fig. 2 x,,x, 20 r=\wii~-~~:ni.~,;i:ii: L~U~;%PUQE~CB~~ The vertices of the basic feasible set are: 1 A(-10) 7 C B (-1, .I) 1 c (0,

0 (0, 0) The value of xl + x2 = .I + .1 = .2 at B 1 =-+O= .14 at A 7 1 =O+-= .I2 at C 8 =O+O=OatO Max x, + x2 = .2 1 i.e. - = 2 V = 5 v

q, = Vx, = 5 ~0.1= 0.5 Fig. 3 q2 = Vx2 = 5 x 0.1 = 0.5

Remark: It is necessary that V be positive, in order to satisfy the non-negativity requirements for the programming variables. If one or more aij I 0, then select a nurnber k with the property that aij -I- k > 0 for all i and j, and add k to every element of the payoff matrix. The value of the modified game is V I- k. Example 9: Solve the game whose pay-off matrix is

Solution,: Let k = 2; then the pay-off matrix becomes

This game is without a saddle point. 2 c Maximum Max 3 4 t Minimax

?'he LPP for paayer B is I Max - = x, + x2 V' I

Subject to xl + 4x2 5 1 J 3x1 + 2x, I1 / 6 XI' X2 2 0 I1 Solving xl + 4x2 = 1 3x, + 2x, = 1 we have xl = 0.2, x2 = 0.2. Rg. 4 1 11 IE Max xl + x2= max I--,-,0, 0.4) max (0.34, 0.25, 4, 0, 0.4) ~0.4 3 4 1 I -= 0.4 I v' I I I; A", 1, Conflict and Co-operntion

Its LPP equivalent for player A is 1 Min 7v = pl + Y2 YI + 3~22 1 4y, + 2~2s1

Min (y, + ~2)= $4

for the point (2.

You can also solve tilo abnve game by graphical method, using the given pay-off matrix. Now you can solve some exardses,

El) Two armies, A and B, wu invulvsd in war game, Each army has available three different strategieq, ~iChPRY-off (abls given bslew, The pay-off table represents square kilometres of Isqcls with ~ositlv~numbers rcsprcsenling gains by A,

Find the saddle point and the value of the game, Socio-Econo~nicEnvironnlent E2) A politician must plan his re-election strategy. He can emphasize jobs or he can emphasize the environment. The voters may be concerned mainly about jobs or about the environment. A pay-off matrix showing the utility of each possible out- come is given below. ,- Voters Jobs Environment

Jobs 25 - 101 Candidate

Environment - 15 , + 30

The political analysts feel that there is a 0.35 chance that the voters will emphasize jobs. What strategy should the candidate adopt? What is the expected utility?

E3) Given table is the pay-off table for the two players. Player B

Strategies 1 2 3

1 Player A 2 3

Determine the optimal strategy for each player by successively eliminating dominated strategies.

E4) The table given below is the pay-off table for two players. Use minimax criterion to solve this game. Player B

Player A 0 : L: -2 -4

E5) Consider the pay-off table for two players given below. By graphical procedure determine the value of the game and the optimal mixed strategy for each player Player B

Player A

E6) Solve the above problem by solving the equivalent linear programming '

%B . a. problems.

.G*' ~ -.>* E7) Obtain a formula for the optimum strategy for the player A in a game that is not. 48 strictly determined (i.e. with no saddle point) having the pay-off matrix given ; below (by graphical method).

Wb - Player B Conflict and Cu-opcmtion

Player A

E8) The pay-off matrix to A is as shown in the table below. Determine the optimum strategy for B. (Use graphical method of solving LPP)

Player B

I Player A 2 3

E9) For the pay-off matrix given below, transform the matrix game into, their corresponding linear prograinming yroble~nsfor player A and player B.

Player A 2 1 -6 8

12.4 GO-OPERATIVE AND NON-COOPERATIVE GAME THEORY

In previous section we have discussed non-cooperative gamc theory. Now we are going to consider games where lhere may be co-operation as well as non-cooperation. There are games which are representatives of many situations in which, two (or more) parties seek to co-ordinate their actions, although they have conflicting prefercnces concerrling which way to co-ordinate. In this context we shall discuss two games (i) Prisoner's Dilemma (ii) Battle of Sexes. The prisoners' dilemma is, perhaps, the most prevalent example of a basic game that recurs in economic context, of course in vastly more complex for~n.Before discussing these two games, we have some examples to illustrate the conflict and cooperation in gamcs. The pay-off matrices of the two games are given .below:

Here two players are A and B having strategies A,, Az and BI, Bz respecdvley. In game I, if the game is communicative, it is gainful for both the players to co-operate, as by either choosing the play (Al, El) or choosing (Az, Bjthey can gain, on average, than if they did not co-operate. -. In game 11, A's gain are matched by B's losses and vice-versa, This is a zero-sum game. Here, the two players are in total conflict as they can not improve their collective pay-off by their collaborative actions. Socio-Economic Environment En prisoners's dilemma and battle of sexes, you will encounter simultaneous conflict and cooperation. Both games represent many economic situations.

12.4.1 "Prisoners' Dilemma" Game We have discussed this game in Example 3. They pay-off matrix is as follows: B's Strategies C R A's Strategies C o(-8) R - 2(-2)

B's outcomes are given in brackets. Here the players (prisoners) are not allowed to communicate as they are in separate cells. Let us recall dominance. Let U be player A's pay-off matrix. If there exists a row k and a row i with UkjI Uij for all values of j except that at least one of value of j is such that Ukj < Uij, we say that the strategy Ak (or kth strategy of A) is dominated by strategy Ai. Observe that the strategy "Confess" dominates "Refuse" for eithcr player. Suppose, that communication between them is allowed, then there is scope of co- operation, as prisoners may feel that they do not want to go to prison for 5 years and this can be avoided if both choose not to confess. This will be the case if they chose (A2, B2), There is also competition between the two players as A may think that he can get away scot-free if he confesses and B does not. Thus, this game involves both conflict and co-operation. In a communicative game for two or more players, some or all of them rnay form coalition. They communicate with each other before making their choices of strategy to decide strategies that will benefit the coalition. Obviously, a player will~onlyjoin the coalition if he gains more utility out of the coalition's share-out than he would get if he played on his own. In the share-out, a player may get more than he would expect to get from the outcome of the game, that is, he 1x1s received a side-paymenl. Coalitions and side-payments are two of the more fascinating aspects of communicative games. The general sort of situation recurs in many contexts in econo~nicswhich can be modelled by the Prisoner's Dilemma game. Consider two firms (producers) selling a similar produc?. Each can advertise, offer items on sale, etc, which may improve its 'own profit and hurt the profits of the rival, holding fixed the actions that its rival may taka. But, increased advertisements by both and so on (like price concession) decrease total net profits. Each of the two firms may resort to an advertisement in order to increase its own profits at the expense of its rivals. When both do so, both do worse than if they could sign a binding agreement to restrict advertising, that is, if they would collude/co-operate. Ofcourse, this is more complex than prisoner's dilemma game since the decisions to be taken is presumably more complex than a simple advertisddo not advertise binary choice. By studying how players might interact, we can gain insight into the rivalry between two firms. For the classic example, you can consider the competition between Boeing company and Air bus company regarding sale of aeroplanes. Consider two countries that are trading partners. Each country can engage on various sorts of protectionist measures that, in some cases, will benefit the protected country, holding fixed the actions of the other country. But if both are engaged in taking protectionist measures, the ovcr all welfare in both countries decreases. Here, again we have the rough character of the prisoner's, dilemma game and insights gained from, say, . ,.. the context of rivalrous oligopolistic, might be transferable to thc context of trade policy. Suppose a group of villages have a piece of common land which can be used for grazing. If every villager in the vicinity lets his cattle use the common land, then the land is over-grazed and becomes a waste land and the cattle do not get sufficient sustenance, Nevertheless, it does not pay any villager to exclude the cattle from the common land as this only benefits the other villagers' cattle. 4"'in A similar example is whaling which is of present day relevance. 12.4.2 "Battle of Sexes" Game Conflict and Co-operation In the second game called 'battle of Sexes', the story concerns a husband and wifc who are trying to decide how to spend an evening. The man would like to watch a cricket match (day-night), the woman wants to see a theatre. But each would rather attend an entertainment with the other than not, and this is more important to each other than attending his or her preferred form of entertainment. The pay-off matrix is as follows: Husband Go to Go to Cricket Theatre Go to Cricket Wife Go to Theatre

Real Examples of Prisoner's Dilemma Game:

We now give two examples modelling tile 'Battle of Sexes' game. In the context of industrial organization, market segmentation among rivals can have this flavour. Suppose two manufacturers of complcrnentary goods contemplate standards to adopt. They may adopt compatible standards or they may, due to various reasons like to adopt different sort of standards. In the context of public finance, two adjacent tax authorities may wish to co-ordinate on the tax system they employ to prevent tax-payers from benefiting from any differences through creative accounting procedures. But if each has a distinct tax clientalc, they may have conflicting preferences over the systems on which they might co-ordinate. In the coiltext of labour economics, both labour union and lnatiagemellt may be better off setting on the others terms, than suffering through a strikc, though each will prcfcr that the other party accommodates itself to the preferred tenns of the first. In the duopoly market there are just two producers, and there may bc competition between thcm andlor there may be co-operation. In an oligopoly there can be competition and/or co-operation between the several producers. The important aspects of both thesc types of market are that the actions of any producer have a perceptible influence on the market behaviour. The distinguishing feature of these two types of market is the interdependence of the actions of the scllers. Let us consider a market with two sellers labelled A and B with corresponding cost functions C1 (ql) and C2 (q2) where qi is the quantity supplied by the it11 seller? The profits ni (q) me given by

Where D-I is ;he inverse of the demand function and is a function of (q, + q2) The inverse de'mnnd function states price as a function of the aggregate quantity sold i.e. p = D-' (ql + q,) where q, and q2 me the levels of duopolists outputs. Suppose the first seller wants to maximize his profit. He can change only ql, but the value he chooses for ql depends on the value of.q,. Same is the situation for the second seller. Thus values chosen for ql and q2 are interdependent and so are the profits that are made. This interdependence is manifested in such a game, The outcome for a player depends not only on his own actions but on the actions of others, The arche type of this situation is: the players try to maximize the points they win (profits here) but the outcome for each player depends not only on his own decisions but also on the decisions of others. In some situations the interests of the players may conflict but in others, it may pay the players to co-operate. This has been illustrated in Prisoner's Dilemma game more elaborately. Socio-Economic Environment 1'Ile theory of slrictly competitive games is not fuju)ly satisfactory as an explanation of oligopoly behaviour. Theboligopolists' interests are not always diametrically opposite, and their behaviour may be chwacterised by a combination of bolh competitive and co- operative elements. The possibility of co-operatioil arises in non-zero (nonconstant) sum games. Such games do not necessarily lead to co-operation, although preferred outcomes rnay be achieved through co-operation. Let us consider the following example. Example 10: Consider thc duopolistic m~rketfor which neither collusive (discussed -- - later) solution is permitted nor bribes and profit redistribution are permitted. Thaa each duopolist has two strategies.

1) he can declare himself as leader and produce a relatively large output or 2) hc car1 declare hinlself a follower and produce ci relatively small output. Once declaration has bccn made he must produce his declared output regardless of what his competitor has declared, A profit matrix is show bclow: Duopolist B Leader Follower

Duopolist A Follower L(1500, 9500) (8000, 8000)

Discuss the outcome of adopting differeni strategies and impact of cooperation and non- cooperalioa.

I Solution: The best outcoine of each duopolist is obtained, if he is a leader and the colnpetitor a follower. The worst happens if their roles are reversed: One night wgue that a sensible strategy for each would be to declare himself a followcr since each would receive a moderately satisfactory profit. If A believes thai B will be a follower, A should be a leader, a fortiori. A similar argument applies to B. Since cach has an incentive to be a leader, their "un-cooperative" behaviour leads each to atkzin his lowest profit level. In fact, by dcminance, the leader strategies of the duopolist constitute an equilibrium pair, whereas the relatively favourable follower strategies do not. Both could gain from co-operative behaviour, but it is not clear that c.ooperation can be achieved successfully. The possibility of finding co-operative solutions depends upon the possibility of nlutually undertaking unbreakable commitments and guarantees. Now you once again look a1 Example 2 discussed in section 12.2. Can you recognize this game? Yes it is a Doopoly, presented as a non-communicative game. Same is the case with Prisoner's Dilcmina g,me presented in example 3 of sec 12.2. Please refer to the games, Duopoly, discussed in the section 12.2 again. Example 11: We presented there "Duopoly" and the 'Prisoncrs' Dilemma' as two examples of non-communicative gmes for two players which are also non zero-sum game.,Discuss the solution of Duopoly game. Solution: From the outcome (payoff) matrjx for the duopoly, game, we see that the strategy of producing 6 units dolninates all other. Here the play (6, 6) is an equilibrium solution, (I1 is infact the Cournot solutipn). Cournot Solution (aftcsr French Economist Augustin Cournot) This is a classical solution of the duopoly problem in which each duopolist maximizes his own quantity with the assumption that the quantity produced by his rival is invariant wit11 respect to his own quantity decision. However, there are other plays which produce greater total profit. (These all lie in the Pareto Optimal set). The set is ((6,0), (6, 2), (6, 4), (4, 2), (4, 4) (4, 6), (2, 4), (2, 6), (10, 6)). The one producing the maximum profit is (4, 4) and this is the solution (Notice that the Cournot solutioll is not Pareto Optimal, in general).

52 h* '. T B'S pgy-off matrix is U , i.e, Conflict and Co-operatin11 -B's Pavoff Matrix B's str~tegies B's strategies

U, + = u, For (6, I)),wo have 257 - 4 = 253

The elements of the set satisfy tlw conditions, Pareto Optimal! A poirit (q:, qi) Is anld to be Pareto Optimal, if for every other point (q,, qJ, either x, (q;, q;) > a, (ql. q2) endfor n2(q;, q,') > n2 (ql. (12). The set of such points 18 the Pareto optimal spt (POS), pach seller wants lo ~vaximizahis profits. Once the producers aie selli~~gquantities (q;, ql), one or other of the producers would dislike a chsngo. This justifies taking n point in PO5 8s an equilibriuln solutlan for a duopoly markol, Collusive solutipll Here duopolists qgree to act in unison to niaxi~oissthe total profit of the industry. The maxirnq~ltolfi1 profit is obtnined by malrimizing tho single function (joint profit),

The point (qy, qi) whiob gives the absolule rnnxirnu~nof this hnction belongs to POS. Nevertheless, this point is not necessarily the natural equilibrium of the duopoly market because one of the producers (ith) may bo getting less profit by makitrg the quantity qt than he would if hetshe were maklng 4 different quantity. Elowever if the duopolists can co-operate, the other duopollst would find it worth while to give a side-payment to overcome [his drawback. This solution to the duopoly market is known as the collusion solution and the duopolists ~sktogather to form a monopoly. Example 12: Discuss the domini~tingsolution for tho Prisoner's Dilemma gane. Pay-off matrices for player A and B are as follows:

Solution: Prom the pay-aff nstricas, we see that the "Confess' strategy dominates the 'refuse' strategy for both A and B. Thus it is in the 'best' interests of both A and B to confess, then both A and B are santenc~dto spend 5 yeas in prison. If, however, they had both adopted the 'refuse' strategy thoy would have only been sentenced to 2 yews of imprisonment. Thus, by pursuing their separate selfish desires they are both worse off. Socio-Ecnnornic Environment Examples 13: Apply dominance in the following non-zero sum game Player B

Consider U. v

Player A z

Solution: For A, x is dominated by y. If A chooses x, u, is better than v for B. Hence once we decide that x will not be played, then we can predict that u will not be played since against y or z, v is better than u. In such a situation y dominates z. Our conclusioa is lhat A will choose y and B, v.

The first step in chains of logic of this sort involves, the application of simple dominance. We go back and forth, first eliminating one or more strategies for one player and then on that basis, eliminating one or more strategies for Lhe others. We arc applying recursive or successive dominances.

Iterative Prisoner's Dilemma Games Here we apply the Prisoner's Dilemma framework to a multiperiod model of behaviour, where the distinction between co-operative and non-cooperative model is blurred. Iterative Prisoner's Dilemma game involves two participants, who are faced with the question whether to stick to a certain course of action or to defects. There is no single best strategy to follow. What is best, depends on the decision of the other player. How a fellow player is likely to act depends on what he expects you to do. The most interesting games involve repeated trials since these games model real situations involving interaction between people or institutions over a period of time. Much has been learnt from computer tournaments between machines which have been programmed to behave and respond to past experiences in a certain way. . (1984), an American political scientist, in a series of experiments using computers to act out Prisoner's Dilemma games (with the computers programmed in many different ways), has found, that the most successful long run strategy is one in which the actors believe that their fellow participants will co-operate. This strategy fails when cheats are encountered and suckers (those who are betrayed) emerge, but it succeeds if grudgers are encountered. The grudger does not seek to defeat the other actor. He does not see the other actor as an opponent but as an accomplice. Successful environments are environments where there are a sufficient number of suckers and grudgers. Professor Axelrod's findings were based on two computer toumaments which took place at the University of Michigan. In repeated Prisoner's Dilemma game (if played many times), equilibrium plays, other than "always confess" will occur. Now we discuss some differences in theory of non-zero sum games when compared to the thcory discussed in zero-sum game. Non-communicative games for two players which are not zero-sum. Duopoly and Prisoner's Dilemma are two examples already discussed. Consider such a game with players A and B with m and n strategies respectively. The problem for A is exactly the same problem as in the zero-sum case and so is equivalent to the linear programming problem. Thus, a solution exists for A and indeed for B. However, B's equivalent 1.p.p. is not the dual of Als 1.p.p. Thus, there are several results like Minimax Theorem etc. that do not generalise in this situation. Also the play-vectors, which give the maximin strategies are not necessarily equilibrium points. This can be illustrated by an example:

-3 1 Let -2 1 - (A2, B2) is the play-vector with maximin strategies, but is not an equilibrium point. The Conflict and Co-operation llorln of rational man mentioned in zero-sum game is riot always satisfactory (The rational man adopts a strategy which miiiimizes the losses he could possibly incur). Another problem in these games is the meaning of mixed strategies, which, however, we will not discuss here. You may try some exercises now:

~10)Apply dominance to find the optimum strategic of A and B from the given pay-off matrix.

Player A y 6 (2) 9 (3) /

El 1) Use dominance property to reduce the following game to 2 x 2 game and hence find the optimal strategies and value of the game. E12) Suppose if the monopoIist sets price p, then the quantity dcn~andedq is given by q = 13 - p. Let C (q) = q + 6.25, i.e. a fixed cost oC 6.25 and il constant marginal cost of 1 per unit. I-Ier profit function is

Show that profit is inilxiinized by producing G units with a profit of 28.75. Suppose a potential entrant believes that if he c;;:ers, the inonopolist will go on making 6 units. Maximizing the profit of the entra~~i,i~nd the quantity to be

produced by him and tlle profits made by the entranl ,II,$ Lhe monopolist. Suppose thc monopolist produces 7 units instead of 6, will the entrant be attracted if he enters provided his proLit is positive. E13) In reccnt literature, the Cournot equilibrium is discussed within the context of game theory wherc thc contribution of Nash is signiricant. In the single-period model each firm has a given set of independent strategies from which it can choose. This concept of strategy is to bc distinguished from a l'irm's best plan of action over several periods when each output decision can be looked upon as a "movc". The 'Cournot-Nash' equilibrium has the characteristics that once a strategy has been chosen, no firm could obtain higher profits by changing its decision. In this sense we have a state of equilibrium Table given below which involves a single-period Prisoner's Dilemma gainc. Single-period pay-off matrix in the Prisoner's Dilemma game Firm 2's choices Hold price constant Cut Price Hold price constant Firms 1's choiccs Cut Price

Examine the game from firm 1's viewpoint in this duopolisdc environment. Givcn that nn identical pay-off matrix applied to firm 2, examine the game from Firm 2's viewpoint also. -- Socio-Econo~l~icEtlvironnmcnt 1.2.5 SOME GENEWALPZATIONS

We IIOW discuss some generalizations oS game theory being pursued presently. One generalization of the garne theory is the Non-Zero-Sum Game, where the sum of the pay-offs to the players need not be zero or any other fixed constant. We have illustrated this by discussing some important games which model many real situations. In such cases it reflects the fact that many competitive situations includc non-competitive aspects that contribute to the mutual advantage or mutual disadvantage of the players. Consider the advertising strategies of competing companies. This can affect not only how they will split the markel but also the total sizes of the market for their competing products. Since mutual gain is possible, non-zero-sum garnes are further classified in terms of the degree to which the players arc pe~nlittedto co-operate. At one extreme is the non- cooperative game, where there is no preplay discussions/communication bctween the players. At the other extreme is the co-operative game, where preplay discussions and binding agreements are permissible. As examples of co-operative game one can consider cornpctitive situat~onsinvolving trade rcgulations between countries, or collective bargaining between labour and management. When there are more than two playcrs, eo-operative garnes also permit some or all of the players to fc~r~ncoalitions. We mention that thc play-vector which give thc maximum strategies are not necessarily thc equilibrium points. Anotller type of imporlant generalization is the n-person game, when more than two players may participate in the game. This genrnlization is needed, because, in many lcind of competitive situations, often there are more than two competitors involved. The convetition among business firms and competition in irlternational diplomacy arc situations which can bc modelled by n-person game. Still another genralization is to the class of infinite games, whcre tlre players have a.n infinite number of pure strategies available to them. T'hese type of games are formulated/designed for the kind of situation, wherc the strategy to be selected can be represented by continuous decision variables. The decision variable might be the time at which to take a certain action or the proportion of one's resources to allocate to a certain activity, in competitive situation. For example, one delivery in cricket or in. baseball can be viewed as such a game. Here, the bowler (or pitcher) can vary his direction and his speed in infinitely many ways. Tlie batsman can vary the position of his bat in infinitely many ways so that both the bowler and the batsman 11ave an infinite r~urnberof strategies. Presently, a lot of research work is going on 'pursuit' and 'evasion' games modelling, how onc missile can catch another, and one nlissile can hit a submarine and how submarine can evade it respectively. - 12.6 SOME AP13LICATIONS AND 1,IMITATIBNS

The general problem of how to make decisions in a cornpctitive environment is a very corninoil and important one. The fundamental conlribution of game theory is that it provides a basic conceptual framework for ibrn~ulatingand analyzing such problems in simple situations. However, there is a considel-able gap between what the theory can handle and the conlplaxity of rnost competitive situations @sing in practice. Therefore, the conceptual tools of game theory usually play just a supplementary role in'tackling these situations. However, due to the importance of the general problem, research work has been going an, with some success, to extend the theory to Inore complex situations. One of the aims of game 1Elec;y was to clarify some problems of economics, but whether it has been successful or not is the subject of controversy among economists. Though it has helped to clarify ideas, but quantitatively it has not, in general, led to any conclusion. However in decision-making, it has been siy~iificantlyhelpful in a quantitalive way. In political studies, the use of game theory has been mainly quantitative, but political scientists are increasingly getting interested in this theory. In parlour games which are mostly too complicated, the ideas of game Jheory; mainly the ideas about the conditions 56 for the fonnntion of coalitions, havr. belpcd to clarify the nature of some of these games

&.b .. I I and to dcvclop new grklncs. Game ~hcoiyhas made nla impict on a surprisingly new Ca~nfiictnnd CQ-op~retinn area- evolution. Here a 'game' ikrvolvrng a genetic slr21egy ahdlor a behavioural strategy is played many times and the optimal choice is likely to survive. Wilh this, we have (lome to thc encl of this wit. Let us take a quick look at what we have covered in this unit.

After going through this unit you have learnt : tj About some sirnple and the various terminologies used in game theory.

2) About two person zero-sum game and also learnt some methods of solving such games, resulti~lgin knowing the optinial strategy of each player and the value of the game. 3) About "prisoncr's dilemma" game and various economic and social situations which 'can be modelled by this g:lme

4) About "Battle of Sexes" game and various economic and social situations which can be modelled by this game. 5. About some generalisations oP the game theory to rnodel complicated co~npetitive situations in the real world. 6. About the limitations of the game theory to model competitive situations.

El) The saddle point is (1, J), and the value of the game is -9. E2) Expected pay-off for the canclidatc are. Job : 2.25 and Environnient : 14. Strategy : Environment E3) For player A, strategy 3 is dominated by strategy 1. Eliminating strategy 3 from further consideration, we have the following pay-off table:

Now for playcr B, strategies 1 and 2 dominates strategy 3. Eliminating this strategy we 'get the reduced payoff matrix as

NOW for player A strategy 1 dominates strategy 2. Hence this yields the pay-off matrix as B

Here, again strategy 2 of I3 is dominated by strategy 1. Player A then receives a pay-off 1 from B and both players use strategy 1. Socio-Economic Environment E4) 1 2 3 Minimum r-- --1

2 0 ~nax(- 3, 0, -4) = 0 3 - 2 -4 i.e. maximin value = 0 L Maximum 5 0 6 Minmax value = 0 Both A and B choose respective strategy 2, The resulting payoff of 0 is the value of the game. Since the value of the game is zero, it is a fair game.

E5) One finds that the third pure strategy of player A is dominated by his second strategy. Here the payoff table can be reduced to the form given below:

Probability I3

For each of the pure strategies available to player B, thc expected pay-off for player A will be

We plot these expected payoff lines on a graph below. For any given value of p, the expected payoff is

Thus, given p, the minimum expected pay-off is given by the corresponding position in the "bottom" line. According to the minimax (or maximin) criterion, player A should select the value p giving the largest minmax expected pay-off. That is Y = V = max {min (- 3 + 5p, 4 - 6p)) - That is, the optimal value of p is given as the intersection of -3 + 5p and 4 - 6p. Conflict and Co-operation i.e. -3+5p = 4-6p

7 4 (p, 1 - p) = (p,, p,) = ( - , - ) is the optimal mixed strategy for player A, , 11 11

The given game does not possess a saddle point. The maximum negative number appearing in the pay-off table in 4. Therefore, we add a constant so that all the elements of the pay-off table become positive. By adding 4 we have the following pay-off tables. (Since maximin value is - 2, the constant to be added should be greater than - 2). p 2 q2

The LPP for player A is as follows: Minimize x* = x, + x2 Subject to 4x1 + 9x2 2 1 2xJ + 8x2 2 1

xi 2 0 i=1,2 xl+x2=1 The LPP for player B is as follows: Maximize = y* = yl + y2 + y3 Subject to 4y1 + 2y2 + 6y3 5 1

~YIf 8~2 + Y, 5 1 yi2 0, i = 1, 2, 3, yl.+ y2 4 yj = 1

The expected pay-off for A, assuming that B plays strategy 1 is El. E,(P) = a11 P + a21 (1 - P) Also E2(p) = a12 P + a22 (1 - P) The optimum strategy for player A (p, I - p) is found by letting E, = E2 i.e., "11P+a21(1-~)=a,,P+a22(1-~)

a,, - a,, - a,, + %, + 0 since the game is not strictly determined. The optimum strategy for player B is (q, 1 - q) where Socio-Economic Environment The value of the game is

91 q2 E8) Let -=v yl,-=v Y 2 Desired LPP is:

Subject to ZY, + 5~25 1

,Solving this graphically.

15 13 Adding - = - , that is v = - V 13 5

Pip, 7

E9) Adding C = 10. the pay-off matrix is modified to

The LPP problem for A is Minimize z = x, + x2 + x3 .

Subject to . .

E10) Player A will not choose strategy x because whichever stratqy player B picks, strategy y gives player A a higher pay-off. Since v dominates u after x is removed. from consideration, B wilI not choose strategy u, that is, B will choose strategy v. If A makes a similar prediction based on the assumption that B makes the assumption atiove, then strategy y dominates z, and we can presllme that z won't be played. Hence, we conclude that A .will choose the strategy, y and B, the strategy v. ~11)Obviously, the third column is dominated by the first column. Hence third column Conflict and Ca-operation can be deleted. Modified pay-off matrix is

Now, we observe that in this vatrix, no row (or column) dominates another row (ar column). But we notice thqt the third row dominates a convex linear combisatinn of first and socand rows, since

Thus, one cap delete sither of the first and second rows by modified dominance. Deleting the first row we have the modified pay-off matrix as

Now Use E7 ta get the final answers,

and n = 12 x 6 - 36 - 6.25 = 36 - 6.25 = 29.75 If the (now) entra~~tproduces ql units, then the pric~of each unit becomes 13 - 6 - ell and the net profit for this entrant is

The maximization of Illis profit gives q; = 3 with net profit of 2.75. Since this is positlye, the entrant enters, Profit of [he monopolist is (13 -6-3). 6-6-625

If the monopolisL is producing. q* = 7 units, and Ihe entrant believes that if he enters, tile monopolist will go on producing 7 units, the net profit is (13 - 7 - q,) q,- q, -6.25. Maximizing we get ql* = 512 with net profit zero. So the entrant has no iqclination Io enter, E13) From th~table, the best aption available to firm 1 is to cut its price. Since Firm 2 has an identical pay-off matrix, it will also cut its price. From the table one can see that vut pricelcut price option is the third best outcome and yet it can not be improved upon ~nlessthere is collusion. Each firm will consider, in retrospect, that it could not have made n beltcr decision, given the decision of the competing firm. Comparing the option to cut prices with the option to hold prices constant, the pay-off8 for firm 1 are higher whether firm 2 opts to hold price constant or to cut pricas sinca two is greater than zero and minus one is greater than -3.