Game Theory Two Person Games A two person game is any conflict or competition between two people factions countries organizations or businesses We refer to them as the players to simplify the discussion The rules for two person games are really straightforward 1 If one player s loss is equal to the other player s gain in all circumstances then it is called a zero sum game 2 A game defined by a matrix called the game matrix or payoff matrix In this class player I will play rows and player II will play columns A positive entry in a matrix denotes a gain to player I and thus a loss to player II in a zero sum game 3 The game is strictly determined if and only if there is an entry that is the smallest in its row and largest in its column This entry is called the saddle point and is the value of the game A game with a positive value favors player I a negative value favors player II If the value of a game is zero that game is fair 4 The row containing the saddle point if one exists is the best strategy for player I and the column containing the saddle point is the best strategy for player II Identifying the saddle point if any is just a matter of finding the smallest value in each row Then scan up and down to see if any value in its column is higher in value If so the row has no saddle Examples Is the zero sum game represented by the matrix strictly determined If so what is the value of the game Which player has the advantage 2 1 3 2 Yes The 1 is the smallest in its row and largest in its in its column So 1 is the saddle point and value of the game Since it is negative Player I rows is playing a game with a 1 value to him Player II sees a value of 1 to him 2 1 3 0 Yes The zero entry is the smallest in its row and largest in its in its column So zero is the saddle point and value of the game This is a fair game 6 5 7 4 4 3 2 3 3 Yes The 5 entry is the smallest in its row and largest in its in its column So it is the saddle point and value of the game Since it is positive Player I rows is playing a game with a value of 5 to him Player II sees a 5 result 2 1 0 2 Yes The zero entry in the first column is the smallest in its row and largest in 0 1 2 3 its in its column So it is the saddle point and value of the game This is a fair 1 2 1 1 game Arizona State University Department of Mathematics and Statistics 1 of 4 Game Theory In the first three examples above each player had an equal number of strategies In the first two cases 2 each In the third 3 each In the last example Player II columns has 4 strategies while Player I rows has only three The number of strategies doesn t matter to the fairness of the game Only the value at a saddle matters if one exists So far we haven t explained how the game is set up Let s do that now with examples 1 Billy and Mandy are playing a betting game They will each write down one of the numbers 1 2 or three without looking at the other person s choice When they show each other if the sum of the numbers is even Mandy pays Billy that number of pennies When the sum is odd Billy pays Mandy that number of pennies Mandy Billy 1 2 3 a Let Billy be player I Make the matrix that represents this game Keep the numbers in order 1 2 3 2 3 4 3 4 5 4 5 6 The set up is to the left Recall that all even totals favor Billy So the matrix shows positive results for all even totals and negatives for each odd total Mandy is considered in a complementary way in that as we scan a column a negative payoff is taken from Billy and given to her b Is the game strictly determined No A quick scan of rows shows that there is no saddle 2 Suppose Billy and Mandy get tired of this game They decide to change it by letting Bill choose from 1 2 or three However Mandy will choose to write only two or three Billy will get paid on total of 3 Mandy gets paid on 6 and 4 The total of 5 is a push zero payoff Mandy Billy 1 2 3 2 3 3 4 4 0 0 6 a Let Billy be player I Make the matrix that represents this game Keep the numbers in order The set up is the left b Is the game strictly determined No There is a no saddle I think you can guess though that poor Billy is going to get hammered Later we ll see how badly 3 Now Billy really tired of this game He decides to change it by to choose two or three also Mandy will still choose only two or three Billy will get paid on prime totals Mandy Billy 2 3 2 of 4 2 3 4 5 5 6 a Let Billy be player I Make the matrix that represents this game Keep the numbers in order The set up is the left b Is the game strictly determined No There is no saddle Arizona State University Department of Mathematics and Statistics Game Theory Mixed Strategies and Expected Value In these last three examples we probably feel that Billy shouldn t play game two but should be willing to play either one or three However we would like to have some real way of quantifying that feeling We do it in the same way we did with other probability situations We calculate the expected value payoff E of the model Let s call the game matrix A Then let s create a probability row vector matrix P for Player I The vector is conceptually the same as the distribution vector of the Markov process and gives us the proportion of plays we can expect from Player I in each of his strategies We also create a probability column vector matrix Q for player II that describes the proportion of plays we can expect from Player II in each of his strategies It is an exercise in recognition which means it s in the book that expected value is the same as the matrix product Mandy Billy Note that the dimensions are a match so this is a well defined matrix product The result is always a 1 1 matrix which we interpret as a number E Let s calculate the E value for each of our previous games on the assumption that Player I uses his first …