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Analysis of Zero-Sum Games: Strategies, Dominance, and Co-operation

This article explores game theory and the concept of zero-sum games, where one player's gain is another player's loss. Topics covered include dominance analysis, mixed strategies, computing optimal probabilities, co-operative possibilities, and examples of different game classes.

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Analysis of Zero-Sum Games: Strategies, Dominance, and Co-operation

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  1. 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

  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

  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

  5. Zero sum gamesMixed strategies • In this case R2 maximises Rows security level C2 minimises Columns 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

  6. Zero sum gamesMixed strategies • Row introduces a new strategy which is a mix of R1 and R2 • Rmixed: • Choose R1 with probability 4/11 • This increases the security level from 3 to 5.2 • Column introduces a new strategy which is a mix of C1 and C2 • Rmixed: • Choose C1 with probability 5/11 • This improves the security level from 7 to 5.2 • We have created a new Nash equilibrium! Fred Wenstøp

  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

  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

  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

  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

  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

  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

  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

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