Game TheoryTwo-Person GamesA two-person game is any conflict or competition between two people, factions, countries, organizations orbusinesses. 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 inits 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 columncontaining the saddle point is the best strategy for player II. Identifying the saddle point if any is just a matter of finding thesmallest value in each row. Then scan up and down to see if anyvalue 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 13 2−  − Yes! The is the smallest in its row and largest in its in its column. So, is1−1−the saddle point and value of the game. Since it is negative, Player I (rows) isplaying a game with a value to him. Player II sees a value of 1 to him.1−2 13 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 74 4 32 3 3     Yes! The 5 entry is the smallest in its row and largest in its in its column. So, itis the saddle point and value of the game. Since it is positive, Player I (rows) isplaying a game with a value of 5 to him. Player II sees a -5 result.2 1 0 20 1 2 31 2 1 1−    − − Yes! The zero entry in the first column is the smallest in its row and largest inits in its column. So, it is the saddle point and value of the game. This is a fairgame.© Arizona State University, Department of Mathematics and Statistics 1 of 4Game TheoryIn the first three examples above each player had an equal number of strategies. In the first two cases, 2 each. Inthe 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 oneexists.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 iseven, Mandy pays Billy that number of pennies. When the sum is odd, Billy pays Mandy that number ofpennies.a. Let Billy be player I. Make the matrix that represents this game.(Keep the numbers in order)The set up is to the left. Recall that all even totals favor Billy. So thematrix shows positive results for all even totals and negatives for eachodd total. Mandy is considered in a complementary way in that as wescan a column, a “negative payoff” is taken from Billy and given toher.b. Is the game strictly determined? No. A quick scan of rows shows thatthere 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. Mandygets paid on 6 and 4. The total of 5 is a push (zero payoff).a. Let Billy be player I. Make the matrix that represents this game. (Keepthe 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 isgoing 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 stillchoose only two or three. Billy will get paid on prime totals!a. Let Billy be player I. Make the matrix that represents this game. (Keepthe numbers in order). The set up is the left.b. Is the game strictly determined? No! There is no saddle. 1 2 31 2 3 42 3 4 53 4 5 6MandyBilly→↓−  − −  − 2 31 3 42 4 03 0 6MandyBilly→↓−  −  − 2 32 4 53 5 6MandyBilly→↓−  − © Arizona State University, Department of Mathematics and Statistics2 of 4Game TheoryMixed Strategies and Expected ValueIn these last three examples, we probably feel that Billy shouldn’t play game two, but should be willing to playeither one or three. However, we would like to have some real way of quantifying that feeling. We do it in thesame 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 isconceptually the same as the distribution vector of the Markov process and gives us the proportion of plays wecan expect from Player I in each of his strategies. We also create a probability column vector (matrix Q) forplayer II that describes the proportion of plays we can expectfrom 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 E PAQ=Note that the dimensions are a match, so this is a well-definedmatrix product. The result is always a matrix which we1 1×interpret as a number, E.Let’s calculate the E value for each of our previous games on theassumption that Player I uses his first (top row) strategy twice asoften as the others. Player II will play a uniform strategy.The calculations for our three examples are to the right.1. For the first matrix example, the calculation shows thatBilly has an expected value of 1. That is a 1 pennyexpected value per play in the long haul. So 100