Game theory GAME THEORY (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Mathematical theory that deals, in an formal, abstract way, with the general features of competitive situations : Like parlor games, military battles, political campaigns, advertising marketing campaigns, etc. Where final outcome depends primarily upon combination of strategies selected by adversaries. Emphasis on decision making processes of adversaries We will focus on simplest case: two person, zero sum games Optimization & Decision 009 9 The odds evens game Player takes evens, player takes odds Each player simultaneously shows or fingers Player wins if total of fingers is even loses if it is odd; vice versa for Player Each h player has strategies: t which? h? Payoff table: Player (even) Strategy Player (odd) Optimization & Decision 009 95 Two person, zero sum game Characterized by: Strategies of player ; Strategies of player ; Payoff table. Strategy: predetermined rule that specifies completely how one intends to respond to each possible circumstance at each stage of game Payoff table: shows gain (positive or negative) for one player that would result from each combination of strategies for the players. Optimization & Decision 009 96 Game theory Prototype example Primary objective is development of rational criteria for selecting a strategy. Two key assumptions are made:. Both players are rational;. Both players choose their strategies solely to promote their own welfare ( no compassion for opponent). Contrasts with decision analysis, where assumption is that decision maker is playing a game with passive opponent nature which chooses its strategies in some rom fashion. Two polititians running against each other for senate Campaign plans must be made for final days Both polititians want to campaign in key cities Spend either full day in each city or full days in one Campaign manager in each city assesses impact of possible combinations for polititian his opponent Polititian shall use information to choose his best strategy on how to use the days Optimization & Decision 009 97 Optimization & Decision 009 98
Formulation Identify the players, the strategies of each player the payoff table Each player has strategies:. Spend day in each city. Spend days in Bigtown. Spend days in Megalopolis Appropriate entries for payoff table for politician are total net votes won from the opponent resulting from days of campaigning. Optimization & Decision 009 99 Variation of example Given the payoff table, which strategy should each player select? Politician Total Net Votes Won by Politician (in Units of,000 Votes) Politician Strategy 0 5 0 Apply concept of dominated strategies to rule out succession of inferior strategies until only choice remains. Optimization & Decision 009 500 Dominated strategy A strategy is dominated by a second strategy if the second strategy is always at least as good ( 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. For player, strategy is dominated by strategy. Resulting reduced table: 0 5 Optimization & Decision 009 50 Variation of example (cont.) Strategy for player is now dominated by strategies of player. Reduced table: 0 Strategy of player dominated by strategy. Reduced table: Strategy for player dominated by strategy. Both players should select their strategy. Optimization & Decision 009 50 Value of the game Payoff to player when both players play optimally is value of the game. Game with value of zero is a fair game. Concept of dominated strategy is useful for: Reducing size of payoff table to be considered; Identifying optimal solution of the game (special cases). Optimization & Decision 009 50 Variation of example Given the payoff table, which strategy should each player select? Saddle point Total Net Votes Won by Politician (equilibrium (in Units of,000 Votes) solution) Politician Strategy 6 0 5 Maximum 5 0 6 Minimax value Politician Minimum 0 Maxmin value Optimization & Decision 009 50 Both politicians break even: fair game!
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 should choose the one whose maximum payoff to player is the smallest. Optimization & Decision 009 505 Variation of example Given the payoff table, which strategy should each player select? Total Net Votes Won by Politician (in Units of,000 Votes) Cycle! Unstable solution Politician Strategy Minimum 0 Maxmin value Politician 5 Maximum 5 Minimax value Optimization & Decision 009 506 Variation 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 rom basis. 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 =,,..., m) y j = probability that player will use strategy j ( j =,,..., n) Probabilities need to be nonnegative add to. These plans (x, x,..., x m ) (y, y,..., y n ) are usually referred to as mixed strategies, the original strategies are called pure strategies. Optimization & Decision 009 507 Optimization & Decision 009 508 When the game is actually played Expected payoff It is necessary for each player to use one of her pure strategies. Pure strategy would be chosen by using some rom device to obtain a rom observation from the probability distribution specified by the mixed strategy. This observation would indicate which particular pure strategy to use. Suppose politicians select the mixed strategies (x, x, x ) = (½,½,0) (y, y, y ) = (0,½,½). 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 = pxy ij i j i= j= p ij is payoff if player uses pure strategy i player uses pure strategy j. Optimization & Decision 009 509 Optimization & Decision 009 50
Expected payoff (cont.) possible payoffs,,,, each with probability ¼ Expected payoff is ¼ =¼ 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 is played many times Game theory extends the concept of the minimax criterion to games that lack a saddle point thus need mixed strategies Minimax criterion for mixed strategies A given player should select the mixed strategy that maximizes the minimum expected payoff to the player Optimal mixed strategy for player is the one that provides the guarantee (minimum expected payoff) that is best (maximal). Value of best guarantee is the maximin value ν Optimal strategy for player provides the best (minimal) guarantee (maximum expected loss) Value of best guarantee is the minimax valueν Optimization & Decision 009 5 Optimization & Decision 009 5 Stable unstable solutions Minimax theorem Using only pure strategies, games not having a saddle point turned out to be unstable because ν < ν Players wanted to change their strategies to improve their positions For games with mixed strategies, it is necessary that ν = ν for optimal solution to be stable This condition always holds for such games according to the minimax theorem of game theory Minimax theorem: If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimax criterion 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. But how to find the optimal mixed strategy for each player? Optimization & Decision 009 5 Optimization & Decision 009 5 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 ) x = x, so it is necessary to solve only for the optimal value of x Plot expected payoff as a function of x for each of her opponent s pure strategies Then identify: point that maximizes the minimum expected payoff opponent s minimax mixed strategy Optimization & Decision 009 55 Back to variation of example Politician Politician Probability y y y Probability Pure strategy x 0 x 5 For each of the pure strategies available to player, the expected payoff for player is (y,y,y ) Expected payoff (,0,0) (0,,0) (0,0,) 0x + 5( x ) = 5 5x x + ( x ) = 6x x ( x ) = + 5x Optimization & Decision 009 56
Optimal solution for politician Optimal solution for politician ν = ν = max{min{ + 5 x, 6 x}} 0 x * x = 7 * x = ν = ν = Minimum expected payoff Optimization & Decision 009 57 Expected payoff resulting from optimal strategy for all values of x satisfies: y(5 5x) + y ( 6x) + y( 5x) ν = ν = When player is playing optimally, x 7 = 0 y + y + y = ν = Also y + y + y = So y = 0, y 5 6 = y = Optimization & Decision 009 58 Other situation Solving by linear programming If there should happen to be more than two lines passing through the maximin point, so that more than two of the y j* values can be greater than zero, this condition would imply that there are many ties for the optimal mixed strategy for player. Set all but two of these y j* values equal to zero solve for the remaining two in the manner just described. For the remaining two, the associated lines must have positive slope in one case negative slope in the other. Any game with mixed strategies can be transformed to a linear programming problem applying the minimax theorem using the definitions of maximin value ν minimax value ν. Define ν = xm+ = yn+ Optimization & Decision 009 59 Optimization & Decision 009 50 LP problem for player Maximize x subject to... m+ p x + p x +... + p x x m m m+ p x + p x +... + p x x m m m+ p x + p x +... + p x x n n mn m m+ x + x +... + x = x 0 for i=,,..., m i Optimization & Decision 009 5 m LP problem for player Minimize y subject to... n+ p y + p y +... + p y y 0 n n n+ p y + p y +... + p y y 0 n n n+ p y + p y +... + p y y 0 m m mn n n+ y + y +... + y = y 0 for j=,,..., n j Optimization & Decision 009 5 n 5
Duality Player LP problem 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 (show it ) Still a loose end What to do about x m+ y n+ being unrestricted in sign in the LP formulations? If ν 0, add nonnegativity constraints If ν 0, either:. Replace variable without a nonnegativity constraint by the difference of two nonnegative variables;. Reverse players so that payoff table would be rewritten as the payoff to the original player. Add a sufficiently large fixed constant to all entries in payoff table that new value of game will be positive Optimization & Decision 009 5 Optimization & Decision 009 5 Example Consider again variation after dominated strategy for player is eliminated Adding x 0 yields x = 7, x =, x = Dual problem yields (y 0) y = 0, y 5 6 =, y =, * y = Maximize x subject to 5x x x + x x x x x x+ x = x, x Optimization & Decision 009 55 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. Optimization & Decision 009 56 Extensions (cont.) 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 preplay discussions binding agreements are permitted Infinite games: players have infinite number of pure strategies available to them. Strategy to be selected can be represented by a continuous decision variable Conclusions General problem of how to make decisions in a competitive environment is a very common important one Fundamental contribution of game theory is a basic conceptual framework for formulating analyzing such problems in simple situations Research is continuing with some success to extend the theory to more complex situations Optimization & Decision 009 57 Optimization & Decision 009 58 6