What is your optimal mixed strategy? What is your opponent’s? How
much should you expect to win or lose on average?- The following payoff table offers a simple depiction of the strategy
choices of the Allies and Germany with respect to the 1944 D-Day
invasion during World War II. The Allies can land at either Calais or
Normandy, and Germany can mount a defense at one, but not both,
locations. Payoffs can be interpreted as the Allies’ probability of
ultimately winning the war.
Find the mixed-strategy equilibrium. Explain briefly these optimal
strategies. What is the value of the game, that is, the Allies’ winning chances?
444 Chapter 10 Game Theory and Competitive StrategyGermany
Calais Normandy
Calais .6 .9
Allies
Normandy .8 .6- In the game show Jeopardy, Bob with $10,000 and Dan with $6,000 are about
to place their bets in Final Jeopardy. (The third player has so little money
that he cannot possible win.) Each secretly places his bet and then answers
a final question, winning his bet with a correct answer, losing it if he is
wrong. Both know that either’s chance of a correct answer is .5 (and these
chances are independent). After answers are given and bets are added or
subtracted, the person with the most total money wins (keeping his money
and returning to play again the next day). The loser gets nothing.
This situation is equivalent to a zero-sum game, where Bob seeks to
maximize his chance of winning (and Dan wants to minimize Bob’s
chance). As shown in Table A, Bob’s strategic options are to make a shut-
out bid, $2,001, giving him an unbeatable $12,001 if he answers correctly,
or to bid nothing, $0. Dan’s options are to bid his entire winnings,
$6,000, or to bid nothing, $0.
a In Table A, supply Bob’s winning chances for the two missing entries.
(For example, the lower-left entry shows that if Bob doesn’t bet but
Dan does, Bob’s winning chance is .5—i.e., when Dan answers
incorrectly.) Then, determine both players’ equilibrium strategies and
the value of the game (i.e., Bob’s winning chances). Does either
player use a mixed strategy?
Table A
Dan
Bob Bet $6,000 Bet $0
Bet $2,001 ___ ___
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