Part 1 Nocooperative Equilibria in Normal Form Games

Part1-3 Some Classical Case in Game Theory

minimax method for zero-sum game

• Maximin： In a zero-sum game, for the player whose

strategies are arrayed along the rows, this is the maximum over the rows of the minimum of his payoffs across the columns in each row. 參賽者認為「任何情況下，我的對手總是選擇讓我報酬最小的策略」，因此「我的最適反應，自其中選擇最大報酬的策略」

• Minimax： In a zero-sum, for the player whose strategies

are arrayed along the columns, this is the minimum over the columns of the maximum of his payoffs across the rows in each column.

參賽者認為「任何情況下，我的對手總是選擇讓他報酬最大的策略」，因此「我的最適反應，自其中選擇讓他報酬最小的策略」

• Minimax method： Searching for a NE in a zero-sum game by finding

the maximin and the minimax and seeing if they are equal and attained for the same row-column pair.

So, minimax （ maximin ） method still a type of best-response calculation and it is appropriate for finding the NE.

That is, my opponent’s minimax strategy is his best response to my maximin strategy and vice versa.

• this method may seem extremely pessimistic and conservative.

零合賽局下： Maximin strategy equilibrium 幾乎等同 NE 。若零合賽局，且只有一 NE ，按maximin rule 即可找到。

非定合賽局中， NE 與 Maximin solution 之解不必然相同

• 。 NE ：（用力，用力）、（不用力，不用力）• 。 Maximin solution ：（不用力，不用力）

• 。 NE ：（ go, wait ）、（ wait, go ）• 。 Maximin solution ：（ wait, wait ）

• remarks of maximin solution Maximin strategy is extremely cautious strategy. 1. solution concept 2. refinement ： risk dominant strategy, payoff dominant strategy

risk dominant strategy• 3. how much caution is rational ？ trembling-hand equilibrium

the Battle of the Sexes

一致性選擇相對不一致性選擇，對個別及整體均較好

Chicken game

• 一致性選擇對參賽者而言本身即為一「困境」

Hawk vs. Dove

賽局在生物學的應用