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Two Player Zero Sum Games

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  1. Two Player Zero Sum Games Virtual Material for Statistics 802

  2. The General (m by n) Two Player, Zero Sum Game • 2 players • opposite interests (zero sum) • communication does not matter • binding agreements do not make sense

  3. The General Two Player Zero Sum Game • Row has m strategies • Column has n strategies • Row and column select a strategy simultaneously • The outcome (payoff to each player) is a function of the strategy selected by row and the strategy by column • The sum of the payoffs is zero (zero-sum, remember)

  4. Sample Game Matrix • Column pays row the amount in the cell • Negative numbers mean row pays column

  5. 2 by 2 Example • Row collects some amount between 14 and 67 from column in this unfair game (The game is unfair because column can not win.)

  6. 2 by 2 Example Row Interchange • Rows, columns or both can be interchanged without changing the structure of the game. In the two games below Rows 1 and 2 have been interchanged but the games are identical!!

  7. 2 by 2 representations (Each player has 2 strategies) • Each 2 by 2 game has 4 representations • original • interchange rows • interchange columns • interchange rows and columns

  8. Simple Games - #1Row’s choice Reminder: Column pays row the amount in the chosen cell. You are row. Should you select row 1 or row 2 and why? Remember, row and column select simultaneously.

  9. Simple Games - #1Row’s Answer Reminder: Column pays row the amount in the chosen cell. You are row. Should you select row 1 or row 2 and why? Remember, row and column select simultaneously. You should select row 2 because regardless of which column is chosen row 2 is better. If column selects col 1 then row 2 yields $34 instead of only $11 while if column selects col 2 row 2 yields $42 instead of only $27. Row wants to collect as much as possible.

  10. Simple Games - #1Column’s choice Reminder: Column pays row the amount in the chosen cell. You are column. Should you select col 1 or col 2 and why? Remember, row and column select simultaneously.

  11. Simple Games - #1Column’s Answer Reminder: Column pays row the amount in the chosen cell. You are column. Should you select col 1 or col 2 and why? Remember, row and column select simultaneously. You should select col 1 because regardless of which row is chosen col 1 is better. If row selects row 1 then col 1 pays only $11 instead of $27 while if row selects row 2 col 2 pays only $34 instead of $42. Column wants to pay as little as possible.

  12. Domination Reminder: Column pays row the amount in the chosen cell. We say that row 2 dominates row 1 since row 2 is better regardless of which column is chosen. Similarly, we say that column 1 dominates column 2 since each outcome in column 1 is better than the corresponding outcome in column 2.

  13. Domination Reminder: Column pays row the amount in the chosen cell. We can always eliminate rows or columns which are dominated in a zero sum game.

  14. Simple Games - #1Game Solution Reminder: Column pays row the amount in the chosen cell. Thus, we have solved our first game (and without using DS for Windows.) Row will select row 2, Column will select col 1 and column will pay row $34. We say the value of the game is $34. We previously had said that this game is unfair because row always wins. To make the game fair, row should pay column $34 for the opportunity to play this game.

  15. Simple games - #2 Answer the following 3 questions before going to the following slides. What should row do? (easy question) What should column do? (not quite as easy) What is the value of the game (easy if you got the other 2 questions)

  16. Simple games - #2Row’s choice As was the case before, row should select row 2 because it is better than row 1 regardless of which column is chosen. That is, $55 is better than $18 and $30 is better than $24.

  17. Simple games - #2Column’s choice Until now, we have found that one row or one column dominates another. At this point though we have a problem because $18 < $24 But $55 > $30 Therefore, neither column dominates the other.

  18. Simple games - #2Column’s choice – cont. However, when column examines this game, column knows that row is going to select row 2. Therefore, column’s only real choice is between paying $55 and paying $30. Column will select col 2, and lose $30 to row in this game. Notice the “you know, I know” logic.

  19. Simple games - #3 Answer the following 3 questions before going to the following slides. What should row do? (difficult question) What should column do? (difficult question) What is the value of the game (double difficult question since the first two questions are difficult)

  20. Simple games - #3 This game has no dominant row nor does it have a dominant column. Thus, we have no straightforward answer to this problem.

  21. Simple games - #3Row’s conservative approach Row could take the following conservative approach to this problem. Row could look at the worst that can happen in either row. That is, if row selects row 1, row may end up winning only $25 whereas if row selects row 2 row may end up winning only $14. Therefore, row prefers row 1 because the worst case ($25) is better than the worst case ($14) for row 2.

  22. Simple games - #3Maximin Since $25 is the best of the worst or maximum of the minima it is called the maximin.

  23. Simple games - #3Column’s conservative way Column could take a similar conservative approach. Column could look at the worst that can happen in either column. That is, if column selects col 1, column may end up paying as much as $34 whereas if column selects col 2 column may end up paying as much as $67. Therefore, column prefers col 1 because the worst case ($34) is better than the worst case ($67) for column 2.

  24. Simple games - #3Minimax Since $34 is the best of the worst or minimum of the maxima for column it is called the minimax.

  25. Simple games - #3Solution ??? When we put row and column’s conservative approaches together we see that row will play row 1, column will play column 2 and the outcome (value) of the game will be that column will pay row $25 (the outcome in row 1, column 1). What is wrong with this outcome?

  26. Simple games - #3Solution ??? What is wrong with this outcome? If row knows that column will select column 1 because column is conservative then row needs to select row 2 and get $34 instead of $25.

  27. Simple games - #3Solution ??? However, if column knows that row will select row 2 because row knows that column is conservative then column needs to select col 2 and pay only $14 instead of $34.

  28. Simple games - #3Solution ??? However, if row knows that column knows that row will select row 2 because row knows that column is conservative and therefore column needs to select col 2 then row must select row 1 and collect $67 instead of $14.

  29. Simple games - #3Solution ??? However, if column knows that row knows that column knows that row will select row 2 because row knows that column is conservative and therefore column needs to select col 2 and that therefore row must select row 1 then column must select col 1 and pay $25 instead of $67 and we are back where we began.

  30. Simple games - #3Solution ??? The structure of this game is different from the structure of the first two examples. They each had only one entry as a solution and in this game we keep cycling around. There is a lesson for this game….. .

  31. Simple games - #3Solution ??? The only way to not let your opponent take advantage of your choice is to not know what your choice is yourself. That is, you must select your strategy randomly. We call this a mixed strategy.

  32. Examination of game 1 Notice that in game 1 (which is trivial to solve) we have that maximin = minimax Minimax maximin

  33. Examination of game 3 Notice that in game 3 (which is hard to solve) we have that maximin < minimax. vhe Value of the game is between maximin, minimax Minimax maximin

  34. Repetition of the game Consider game #3 above . What would you choose • if this game were played only once? • if this game were played many times?

  35. Mixed strategies • Row will pick row 1 with probability p and row 2 with probability (1-p) • Column will pick col 1 with probability q and col 2 with probability (1-q)

  36. Expected values (weighted average) as a function of p

  37. Graph of expected value as a function of row’s mix

  38. Solution • We need to find p to make the expected values against both columns equal • We need to find q to make the expected values against both rows equal

  39. Example - Results Row should play row 1 32% of the time and row 2 68% of the time. Column should play column 1 85% of the time and column 2 15% of the time. On average, column will pay row $31.10.

  40. Example - value of the game • .322*25+.677*34 = 31.097 (Col 1 * row’s mix) • .322*67+.677*14 = 31.097 (Col 2 * row’s mix) • .855*25+.145*67 = 31.097 (Row 1 * col’s mix) • .855*34+.145*14 = 31.097 (Row 2 * col’s mix)

  41. Expect value computation (continued) If row and column each play according to the percentages on the outside then each of the four cells will occur with probabilities as shown in the table

  42. Expect value computation (continued) This leads to an expected value of 25*.276+67*.047+34*.579+14*.098 = 31.097

  43. Solution summary • If maximin=minimax • there is a saddle point (equilibrium) and each player has a pure strategy – plays only one strategy • If maximin does not equal minimax • maximin <= value of game <= minimax • We find mixed strategies • We find the (expected) value or weighted average of the game

  44. Zero-sum Game features • A constant can be added to a zero sum game without affecting the optimal strategies. • A zero sum game can be multiplied by a positive constant without affecting the optimal strategies. • A zero sum game is fair if its value is 0 • A graph can be drawn for a player if the player has only 2 strategies available.