GAME THEORY

Game Theory _ Introduction Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome Finding acceptable, if not optimal, strategies in conflict situations. Abstraction of real complex situation Game theory is highly mathematical Game theory assumes all human interactions can be understood and navigated by presumptions. GAME THEORY

Why Game Theory? All intelligent beings make decisions all the time. AI needs to perform these tasks as a result. Helps us to analyze situations more rationally and formulate an acceptable alternative with respect to circumstance. GAME THEORY

Factors in Game Theory Number of players – If the game involves two person it is called two person game else it is called n-person game. Sum of gains & Losses – If the gains of one player is equal to the losses of the other player then it is called zero-sum game else it is called non-zero sum game. Strategy – The strategy of the player is all possible actions that he will take for every payoff. GAME THEORY

Game Theory Zero Sum Game Non Zero Sum game The sum of the payoffs remains constant during the course of the game. Two sides in conflict Being well informed always helps a player Non Zero Sum game The sum of payoffs is not constant during the course of game play. Players may co-operate or compete Being well informed may harm a player. GAME THEORY

Strategy of Game Theory Pure Strategy It is the decision rule which is always used by the player to select the particular strategy (course of action). Thus each player knows in advance all strategies out of which he selects only one particular strategy. Mixed Strategy Courses of action that are to be selected on a particular occasion with some fixed probability are called mixed strategies are called mixed strategy GAME THEORY

Two Person Zero Sum Game Payoff Matrix Payoff is a quantitative measure of satisfaction a player gets at the end of play. It can be market share, profit, etc. Gain of one person is loss of other person. Thus it is sufficient to construct payoff table for one player only. Each player has available to him a finite no of possible strategies. Player attempts to maximise his gains while player attempts to minimise losses. Decisions are made simultaneously and known to each other. Both players know each other’s payoff’s. GAME THEORY

General Payoff Matrix Player A Strategy Player B strategy B1 B2 . Bn   B1 B2 . Bn A1 a11 a12 a1n A2 a21 a22 a2n Am am1 am2 amn GAME THEORY

Minimax & Maximin Principle Optimal Pure Strategy (Minimax Criterion) To locate the optimal pure strategy for the row player, first circle the row minima: the smallest payoff's) in each row. Then select the largest row minimum. (If there are two or more largest row minima, choose either one.) To locate the optimal pure strategy for the column player, first box the column maxima: the largest payoff in each column. Then select the smallest column maximum. (If there are two or more smallest column maxima, choose either one.) GAME THEORY

Saddle Point Saddle Points; Strictly Determined Games A saddle point is an entry that is simultaneously a row minimum and a column maximum. If a game has one or more saddle points, it is strictly determined. All saddle points will have the same payoff value, called the value of the game. A fair game has a value of zero; otherwise it is unfair, or biased. Choosing the row and column through any saddle point gives optimal strategies for both players under the minimax criterion. GAME THEORY

Example (2 person sum game) Union Strategies Company Strategies   I II III IV Row Min 20 15 12 35 25 14 8 10 40 2 5 4 11 Column Max Maximin = Minimax = Value of Game =12 GAME THEORY

Mixed Game Strategy If pure strategy is both applicable then we have to apply mixed strategy. Both players must determine mixed strategy to optimize their payoff’s. Mixed strategy is evolved using probability. The expected payoff to a player in a game with arbitrary payoff matrix [aij] of order mxn is defined as Where pi are probabilities of player A and qj are probabilities of player B Can be solved using Algebraic, Matrix, Graphical or LP methods GAME THEORY

Rules of Dominance Reduction by Dominance Check whether there is any row in the (remaining) matrix that is dominated by another row (this means that it is ≤ some other row). If there is one, delete it. Check whether there is any column in the (remaining) matrix that is dominated by another column (this means that it is ≥ some other column). If there is one, delete it. Repeat steps 1 and 2 in any order until there are no dominated rows or columns GAME THEORY

Algebraic Method Probability of A player Player A Strategy Player B strategy   B1 B2 . Bn p1 A1 a11 a12 a1n p2 A2 a21 a22 a2n Am am1 am2 amn pn Probability of B player q1 q2 qn GAME THEORY

Algebraic Method GAME THEORY

Algebraic Method The equations shown earlier need to be solved. To do this we first convert the equations as equality. Solve to arrive at the p’s and q’s GAME THEORY

Algebraic Method Player A Player B q1 q2 p1 a11 a12 p2 a21 a22   q1 q2 p1 a11 a12 p2 a21 a22 GAME THEORY

Example Player A Player B B1 B2 B3 B4 A1 3 2 4 A2 A3 A4 8 The example does not have a saddle point. We will apply rules of dominance For player A first row is dominated by third row, hence delete first row In the second matrix, column B1 is dominated by column B3 Player A Player B B1 B2 B3 B4 A2 3 4 2 A3 A4 8 GAME THEORY

Example B2 B3 B4 A2 4 2 A3 A4 8 Now we cannot find any dominant strategies, however the average payoff of B3 & B4 is greater than B2 and hence we may delete column B2 B3 B4 A2 2 4 A3 A4 8 Similarly the average payoff of rows A3 & A4 is better than A2 and hence we can eliminate row A2 GAME THEORY

Example B3 B4 A3 4 A4 8 The problem has been reduced to 2x2 matrix which can be solved using algebraic methods. GAME THEORY

Graphical Method Approach to find solutions for 2 x n or m x 2 games. Let player A have two strategies A1 & A2, and B have n strategies B1, B2,…Bn. For B1 strategy the expected gain for player will be a11p1+a21p2. Similarly for each strategy of B we will have one equation in p1 & p2. Draw the straight lines using these equations. GAME THEORY

Graphical Method The vertical axes will have the strategy of player A and the horizontal axes will have the probability of achievement. The highest point on the lower boundary of these lines will give maximum expected payoff among the minimum expected payoff’s and the optimum value of probability p1 & p2. The m x 2 game is alo treated similarly except that the upper boundary of the straight lines corresponding to B’s expected payoff will give minimum expected payoff. GAME THEORY

Example Player A Player B B1 B2 B3 B4 A1 2 3 -2 A2 4 6 B4 GAME THEORY

LP Method GAME THEORY