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MBA6 Game Theory

MBA6 Game Theory. Fred Wenstøp. Zero Sum Games Introduction. Games where one player wins what the other player loses Co-operation is out of question Conventions: Players: Row and Column Row chooses a row Column chooses a column Choices are made independently and simultaneously

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MBA6 Game Theory

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  1. MBA6 Game Theory Fred Wenstøp

  2. Zero Sum GamesIntroduction • Games where one player wins what the other player loses • Co-operation is out of question • Conventions: • Players: Row and Column • Row chooses a row • Column chooses a column • Choices are made independently and simultaneously • The table shows Row's result • Row wants big numbers • Column wants small numbers • The payoff table is known to both players Fred Wenstøp: MBA6

  3. Row thinks: Column will never choose C1 because it is dominated Therefore it boils down to: Column thinks that Row thinks this Therefore Row will choose R2 since it is dominating The “solution” is R2, C2 Zero Sum GamesAnalysis of Dominance Fred Wenstøp: MBA6

  4. Zero Sum GamesMaximising Security Levels • If there is no dominance: • Find the row that maximises Rows security level (R2) • Find the column that minimises Columns security level (C1) • Saddle point: minimax = maximin • a stable solution since no player can gain by a unilateral move • It is called a Nash equilibrium • But not all games have saddle points... Fred Wenstøp: MBA6

  5. Zero Sum GamesMixed Strategies • In this case R2 maximises Row’s security level C2 minimises Column’s security level • But, • minimax is not equal to maximin • Therefore Row chooses R2 Column had planned to choose C2 • but thinking that Row will choose R2, he chooses C1 instead Row thinks that column thinks this • Therefore, Row chooses R1 Column thinks that Row thinks this.... The solution? Do something unexpected! The game of pure strategies Fred Wenstøp: MBA6

  6. Row introduces a new strategy which is a mix of R1 and R2 Rmixed: Choose R1 with probability 4/11 Rm has increased row’s security level from 3 to 5.2 Likewise: Column introduces a new strategy which is a mix of C1 and C2 Cmixed: Choose C1 with probability 5/11 This improves column’s security level from 7 to 5.2 We have created a new Nash equilibrium! Zero Sum GamesMixed Strategies Fred Wenstøp: MBA6

  7. Mixed StrategiesHow to Compute the Optimal Probabilities • p is the probability that Row will choose R2 • The diagram shows the payoff for all values off p • Depending on whether Column chooses C1 or C2 • Rows maximin and coumns minimax security level is where the lines cross • The line equations: • 2+5p • 9-6p • solution: p = 7/11 C1 C2 Fred Wenstøp: MBA6

  8. Games With Possibilities for Co-operation • The size of the pie depends on both players' moves • This invites to co-operation • Conventions • Both players' result is shown in the table, Row's result first • Several classes of games • Little or now conflict • Games with possibilities for threats • Games with possibilities for coercion • Battle of the sexes games • Prisoners' dilemma games Fred Wenstøp: MBA6

  9. No conflict Both players choose dominant strategies The solution (12;8) dominates all other solutions It is called Pareto optimal since there is no other point where both players would want to move together It is also a Nash point since nobody would leave alone Weak conflict Both players still choose their dominant strategies The solution (12;8) is Pareto optimal It is also a Nash point But it is not dominating since both of them would like to be in another point where they cannot get Little or No Conflict Fred Wenstøp: MBA6

  10. Games with possibilities for threats • The Seller sets the price • The buyer decides on the quantity • If both players choose their dominant strategies, the solution will be • Q=much, P=high • Pareto-optimal • Nash-point • Non-dominating • Buyer can threaten to buy little if Seller does not lower the price • In this case, Seller is vulnerable • New solution: Q=much, P=low • Pareto-optimal • Not a Nash-point • Non-dominating Fred Wenstøp: MBA6

  11. Games with possibilities for coercion • Only Row has a dominant strategy • Apparent natural solution • R1 C1 • Nash • Pareto • Non-dominating • But Row can force Column to choose C2 by selecting R2 • New solution: • R2 C2 • Not Nash • Pareto • Non-dominating Fred Wenstøp: MBA6

  12. Battle of the sexes games • Row and Column both want to introduce a new product • If they both do so, they compete and both will lose • Solutions • No Yes • Yes No • are both Nash points and Pareto points • To get to the right Nash point, it is necessary to signal early Fred Wenstøp: MBA6

  13. Two persons are accused of a serious crime They cannot be convicted unless at least one confesses and turn state evidence against the other If both do this, they will each get 8 years If only one confesses, the other will get 10 years If none confesses, both will be convicted of a minor offence and get 2 years Dominant strategies: Confess Confess Solution: 8 8 Nash point Not at all Pareto Optimal Dominated The prisoners' dilemma Fred Wenstøp: MBA6

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