1. 1

2. 2 You should know by now… u The security level of a strategy for a player is the minimum payoff regardless of what strategy his opponent uses. u A player tries to choose among all strategies available to him, the strategy that maximises the security level. u That is, the option that gives the least worst outcome.

3. 3 You should also know what a saddle point is by now… u A solution (a i, A j ) to a zero-sum two-person game is stable (or in equilibrium) if Player I expecting Player II to Play A j has nothing to gain by deviating from a i AND Player II expecting Player I to Play a i has nothing to gain by deviating from playing A j.

4. 4 Principle II u The players tend to strategy pairs that are in equilibrium, i.e. stable An optimal solution is said to be reached if neither player finds it beneficial to change their strategy.

5. 5 1.3 Saddle Points u Let L denote the best (largest) security level of Player I, and let U denote the best (smallest) security level of Player II. u We shall refer to L as the lower value of the game and to U as the upper value of the game. u If U=L we call this common value the value of the game.

6. 6 1.3.1 Example u L = U A 1 A 2 s i a 1 020 a 2 311 S j 32 L U

7. 7 1.3.2 Example A 1 A 2 A 3 s i a 1 32102 a 2 4151 a 3 8-1-2-2 S j 8210 L U Value of game is 2

8. 8 1.3.1 Theorem u For any zero-sum 2-person game we have L ≤ U. u Proof. u Consider the ith row and jth column of the payoff matrix for some arbitrary choice of i and j. u By definition s i is the smallest element of row i, hence s i ≤ v ij. u Similarly, by definition S j is the largest element in column j, hence S j ≥ v ij. u This implies that u s i ≤ v ij ≤ S j, for all i and j

9. 9 u By definition u L = max s i u = s i for some i and u U = min S j u = S j for some j, u hence L= s i ≤ v ij ≤ S j = U, so L ≤ U. v ij sisi SjSj

10. 10 1.3.1 Lemma (Page 12) u For any zero-sum 2-person game, L = U implies the existence of a pair (i*, j*) such that v i*j* = s i* = S j*. u i.e. There is an entry in the matrix that is both the smallest in its row AND the largest in its column. u Proof: By definition, L= s i for some i, call it i*, and U = S j for some j, call it j*. Hence L= U implies the existence of a pair (i*, j*) such that s i* = S j*. From the definition of min it follows that v i*j* ≥ min {v i*j : j=1,2,...,n} (= s i* ) and from the definition of max we have that v i*j* ≤ max {v ij* :i=1,2,...,m} (=S j* )

11. 11 We therefore conclude that s i* ≤ v i*j* ≤ S j* But since we already established that s i* = S j* we conclude that v i*j* = s i* = S j*.

12. 12 1.3.1 Definition: Saddle Point (Page 13) u An entry (i*, j*) of the payoff matrix is said to be a saddle point iff v i*j* = s i * = S j*. u I.e. A saddle point is BOTH the smallest in its row and the largest in its column.

13. 13 u 1.3.2 Theorem: For any 2-person zero sum game, the lower value of the game is equal to the upper value of the game if and only if the payoff matrix possesses a saddle point. u Proof. u Necessity (L=U implies the existence of a saddle point) is provided by Lemma 1.3.1.

14. 14 Sufficiency (existence of a saddle point implies that L=U): Assume that there is a saddle point, say v i*j*. By definition then, S j* = v i*j* = s i*. Since by definition L ≥ s i* and U ≤ S j*, U ≤ S j* = v i*j* = s i* ≤ L But Theorem 1.3.1 claims that U ≥ L. Hence it follows that U = L.

15. 15 Saddle Point

16. 16 Summary u If the players follow the two Principles (Best Security Level and Equilibrium) and the payoff matrix has a saddle point, then there is a pair of pure strategies (one for each player) which is a stable solution to the game. This solution is given by the saddle point. u When we say a pure strategy we mean the player uses one row (or one column) all the time.

17. 17  Solve the 2–person zero–sum game whose payoff matrix is below. ie. Find saddle points, if any. Find the value of the game. State the strategies the players should use, based on the philosophy given earlier. u See lecture for solution. Example a1a1 a2a2 a3a3 a4a4 A1A1 A2A2 A3A3 A4A4

18. 18 Example u For the two-person zero-sum game whose payoff matrix is given below, find the values of x for which there is a saddle point. Solve the game for these values of x. u See lecture for solution.

19. 19 Two–person Constant–sum Games u A two–person constant–sum game is a two player game in which, for any choice of both players’ strategies, the row player’s payoff and the column player’s payoff add up to a constant value, c. u A two–person zero–sum game is a special case of this. u A two–person constant–sum can be approached in the same way as a two– person zero–sum game.

20. 20 Example: In a certain time slot, two TV networks are vying for 100 million viewers. They each have the same three choices for that time slot. Surveys suggest the following numbers of viewers would tune in to each network (in millions). Network 2 Network 1 Western Soap Opera Comedy Western Soap Opera Comedy (35,65) (45,55) (38,62) (15,85) (58,42) (14,86) (60,40) (50,50) (70,30)

21. 21 u As with zero–sum games we could just enter the payoffs to Player 1, with the understanding that Player 2 gets (100 – Player 1’s payoff) u By subtracting 50 from all entries we can convert this to a zero–sum game. u In general by subtracting c/2, a two–person constant–sum game (where c is the constant sum) can be converted to a two–person zero–sum game and thus the same ideas can be used.

22. 22 Question: u What happens if we do not have a saddle point?

23. 23 So, what do we do? We cannot guarantee the existence of a solution satisfying both principles One idea in this case is to think of playing the game repeatedly and looking at expected payoffs, rather than the actual payoff on any one play of the game.

24. 24 The BIG Fix Mixed Strategies Mixed Strategies Each player will mix his/her decisions using some probability distribution. Thus on one play of the game Player 1 may use strategy a 2, on the next play a 4, then a 2, then a 2, then a 1, ??? but ….

25. 25 New Game Player 2 Player 1 Payoff Table Randomise your decisions, mate! How should I randomise?