BRIEF INTRODUCTION TO GAME THEORY Game Theory Mathematical theory that deals with the general features of competitive situations. Examples: parlor games, military battles, political campaigns, advertising and marketing campaigns, etc. Final outcome depends primarily upon combination of strategies selected by adversaries; emphasis on decision-making processes of adversaries. Applications: business, economics, etc. We will focus on the simplest case: two-person, zero-sum games 572
Two-person, zero-sum game Example: The odds and evens game Each player simultaneously shows or 2 fingers. One player takes evens, the other player takes odds. Player wins the bet (say $) if total of fingers is even and loses (say $) if it is odd; vice-versa for Player 2. Each player has 2 strategies: show or 2 fingers. Payoff table: Player (even) Strategy Player 2 (odd) 2-2 - 573 Two-person, zero-sum game Characterized by: Strategies of player ; Strategies of player 2; Payoff table. Strategy:predetermined rule that specifies completely how one intends to respond to each possible circumstance at each stage of the game. Payoff table: shows gain (positive or negative) for one player that would result from each combination of strategies for the two players. 574 2
Game theory Primary objective is the development of rational criteria for selecting a strategy. Two key assumptions are made:. Both players are rational; 2. Both players choose their strategies solely to promote their own welfare(no compassion for opponent). Contrasts with decision analysis, where it is assumed that the decision maker is playing a game with passive opponent nature which chooses its strategies in a random fashion. 575 Prototype example Two politicians are running against each other for Senate. Campaign plans must be made for final two days. Both politicians can do campaign in two key cities, Bigtownand Megalopolis, spending either one full day in each city, or two full days in one city. Campaign manager in each city assesses the impact of possible combinations for politician and his opponent. Politician shall use information to choose his best strategy on how to use the two days. 576 3
Formulation Identify the two players, the strategies of each player and the payoff table. Each player has three strategies:. Spend one day in each city. 2. Spend two days in Bigtown. 3. Spend two days in Megalopolis. Appropriate entries for payoff table for one politician are the total net votes wonfrom the opponent resulting from the two days of campaigning. 577 Formulation Strategy Politician 2 3 Total Net Votes Won by Politician (in Units of,000 Votes) Politician 2 2 3 Form of payoff table for politician for the political campaign problem. 578 4
Variation of example Player2 Strategy 2 3 2 4 Player 2 0 5 3 0 - Which strategy should each player select? Apply concept of dominated strategies to rule out succession of inferior strategies until only one choice remains. 579 Dominated strategy A strategy is dominatedby a second strategy if the second strategy is always at least as good (and sometimes better) regardless of what the opponent does. A dominated strategy can be eliminated immediately from further consideration. Payoff table includes no dominated strategies for player 2. For player, strategy 3 is dominated by strategy. Resulting reduced table: 2 3 2 4 2 0 5 580 5
Variation of example (cont.) As both players are rational, strategy 3 for player 2 is now dominated by strategies and 2 of player. Reduced table: Strategy 2 of player is now dominated by strategy. Reduced table: 2 2 2 0 2 2 Strategy 2 for player 2 dominated by strategy. Both players should select their strategy. 58 Value of the game Intheexample, politician receivesa payoff000 votes from politician 2. Payoff to player when both players play optimally is the value of the game. Game with value of zero is said to be a fair game. Concept of dominated strategy is useful for: Reducing size of payoff table to be considered. Identifying optimal solution of the game in special cases as the variation of the example. 582 6
Variation 2 of example Given the payoff table, which strategy should each player select? No dominated strategies! Player2 Saddle point (equilibrium solution) Strategy 2 3-3 -2 6 Player 2 2 0 2 3 5-2 -4 Maximum 5 0 6 Minimax value Minimum -3 0 Maximin value -4 Both politicians breakeven: fair game! 583 Minimax criterion Each player should play in such a way as to minimize his maximum losses whenever the resulting choice of strategy cannot be exploited by the opponent to then improve his position. Select a strategy that would be best even if the selection were being announced to the opponent before the opponent chooses a strategy. Player should select the strategy whose minimum payoff is largest, whereas player 2 should choose the one whose maximum payoff to player is the smallest. 584 7
Variation 3 of example Given the payoff table, which strategy should each player select? Player Strategy Player2 2 3 0-2 2 2 5 4-3 3 2 3-4 Maximum 5 4 2 Minimum -2 Maximin value -3-4 Minimax value Cycle! Unstable solution 585 Variation 3 of example (cont.) Originally suggested solution is an unstable solution (no saddle point). Whenever one player s strategy is predictable, the opponent can take advantage of this information to improve his position. An essential feature of a rational plan for playing a game such as this one is that neither player should be able to deduce which strategy the other will use. It is necessary to choose among alternative acceptable strategies on some kind of random basis. 586 8
Games with mixed strategies Whenever a game does not possess a saddle point, game theory advises each player to assign a probability distribution over her set of strategies. Let: x i = probability that player will use strategy i (i =,2,, m) y j = probability that player 2 will use strategy j (j =,2,, n) Probabilities need to be nonnegative and add to. These plans (x, x 2,..., x m )and (y, y 2,..., y n )are usually referred to as mixed strategies, and the original strategies are called pure strategies. 587 When the game is actually played It is necessary for each player to use one of her pure strategies. Pure strategy would be chosen by using some random device to obtain a random observation from the probability distribution specified by the mixed strategy. This observation would indicate which particular pure strategy to use. 588 9
Expected payoff Suppose politicians and 2 select the mixed strategies (x, x 2, x 3 ) = (0.5,0.5,0)and (y, y 2, y 3 ) = (0,0.5,0.5). Each player could then flip a coin to determine which of his two acceptable pure strategies he will actually use. Useful measure of performance is expected payoff: Expected payoff for player = m n i= j= p x y ij i j p ij is payoff if player uses pure strategy iand player 2 uses pure strategy j. 589 Expected payoff(cont.) 4 possible payoffs ( 2,2,4, 3), each with probability 0.25 Expectedpayoffis 0.25 ( 2+2+4 3) = 0.25 This measure of performance does not disclose anything about the risks involved in playing the game. It indicates what the average payoff will tend to be if the game isplayedmanytimes. Game theory extends the concept of the minimax criterionto games that lack a saddle point and thus need mixed strategies. 590 0
Minimax criterion for mixed strategies A given player should select the mixed strategy that maximizes the minimum expected payoff to the player. Optimalmixed strategy for player is the one that provides the guarantee(minimum expected payoff) that is best(maximal). Value of best guarantee is the maximinvalue ν Optimal strategy for player 2 provides the best (minimal) guarantee(maximum expected loss) Value of best guarantee is the minimaxvalueν 59 Stable and unstable solutions Using only pure strategies, games not having a saddle point turned out to be unstablebecause ν < ν. Players would want to change their strategies to improve their positions. For games with mixed strategies, it is necessary that ν = ν for optimal solution to bestable. This condition always holds for such games according to the minimax theorem of game theory. 592
Minimax theorem Minimaxtheorem:If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimaxcriterion provides a stable solution with ν = ν = ν (the value of the game), so that neither player can do better by unilaterally changing her or his strategy. Mixed strategies becomes quite intuitive if the game is played repeatedly, it requires some interpretation when the game is to be played just once. How to find the optimal mixed strategy for each player? 593 Graphical solution procedure Consider any game with mixed strategies such that, after dominated strategies are eliminated, one of the players has only two pure strategies Mixed strategies are (x, x 2 ) and x 2 = x,so it is necessary to solve only for the optimal value of x. Plot of expected payoff as a function of x for each of her opponent s pure strategies can be done. Plot can be used to identify: point that maximizes the minimum expected payoff; opponent s minimax mixed strategy. 594 2
Backto variation3 ofexample Player Player 2 Probability y y 2 y 3 Probability Pure strategy 2 3 x 0-2 2 - x 2 5 4-3 For each of the pure strategies available to player 2, the expected payoff for player is (y, y 2, y 3 ) Expected payoff (, 0, 0) (0,, 0) (0, 0, ) 0x + 5( - x ) = 5-5x -2x + 4( - x ) = 4-6x 2x - 3( - x ) = -3 + 5x 595 Optimal solution for politician ν = ν = max{min{ 3 + 5 x,4 6 x }} 0 x * x = 7 * x 4 2 = ν = ν = 2 Minimum expected payoff 596 3
Optimal solution for politician 2 Expected payoff resulting from optimal strategy for all values of x satisfies: * * * y (5 5 x) + y2(4 6 x) + y 2 3(3 5 x) ν = ν = When player is playing optimally, x = 7 and 20 * * * y + 2 y2 + 2 y 2 3 = ν = * * * Also y + y2 + y3 = * * * So y = 0, y 5 6 2 = and y 3 = 597 Solving by linear programming Any game with mixed strategies can be transformed to a linear programming problem applying the minimax theorem and using the definitions of maximinvalue ν and minimax value ν. Define ν = xm+ = yn+ 598 4
LP problemfor player Maximize x subject to... and m+ p x + p x +... + p x x 0 2 2 m m m+ p x + p x +... + p x x 0 2 22 2 m2 m m+ p x + p x +... + p x x 0 n 2n 2 mn m m+ x + x +... + x = 2 m x 0 for i=,2,..., m i 599 LP problemfor player2 Minimize y subject to... and n+ p y + p y +... + p y y 0 2 2 n n n+ p y + p y +... + p y y 0 2 22 2 2n n n+ p y + p y +... + p y y 0 m m2 2 mn n n+ y + y +... + y = 2 y 0 for j=,2,..., n j n 600 5
Duality Player 2 LP problem and player LP problem are dual to each other. Optimal mixed strategies for both players can be found by solving only one of the LP problems. Duality provides simple proof of the minimax theorem. 60 Still a loose end What to do about x m+ and y n+ being unrestricted in sign in the LP formulations? If ν 0, add nonnegativity constraints. If ν < 0, either:. Replace variable without a nonnegativityconstraint by the difference of two nonnegative variables; 2. Reverse players and 2 so that payoff table would be rewritten as the payoff to the original player 2; 3. Add a sufficiently large fixed constant to all entries in payoff table that new value of game will be positive. 602 6
Example Consider again variation 3after dominated strategy 3 for player is eliminated. Maximize x subject to and 3 3 5x x 0 2x + 4x x 0 2 3 2x 3x x 0 x x 2 3 + x = 2 0, x 0 2 603 Example Applying the simplex method to this LP problem (after adding the constraint x 3 0) yields: * * * x = 7, x 4 2 2 =, x 3 = Dual problem yields (adding the constraint y 4 0) * * * * y = 0, y 5 6 2 2 =, y3 =, y 4 = 604 7
Extensions Two-person, constant-sum game: sum of payoffs to two players is fixed constant (positive or negative) regardless of combination of strategies selected. N-person game,e.g., competition among business firms, international diplomacy, etc. Nonzero-sum game: e.g., advertising strategies of competing companies can affect not only how they will split the market but also the total size of the market for their competing products. Size of mutual gain (or loss) for the players depends on combination of strategies chosen. 605 Extensions Nonzero-sum games classified in terms of the degree to which the players are permitted to cooperate. Noncooperative game: there is no preplay communication between players. Cooperative game: where preplaydiscussions and binding agreements are permitted. Ex: trade regulations between countries, collective bargaining between labor and management. Infinite games: players have infinite number of pure strategies available to them. Strategy to be selected can be represented by a continuous decision variable. 606 8