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Strategy14Handout

# Strategy14Handout - Your Choice of Gambles A1= A2=.2.8.8.2...

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10/23/2008 1 Your Choice of Gambles A1= B1= A2= B2= .8 \$4000 .2 \$0 1 \$3000 0 \$0 .2 \$4000 .8 \$0 .25 \$3000 .75 \$0 It turns out that if you preferred A1 to B1, then you should prefer A2 to B2. One way to see this… Your Choice of Gambles A1= B1= A2= B2= .8 \$4000 .2 \$0 1 \$3000 0 \$0 .2 \$4000 .8 \$0 .25 \$3000 .75 \$0 A1= B1= .75 \$0 .25 A2= .75 \$0 .25 B2= Your Choice of Gambles A1= B1= A2= B2= .8 \$4000 .2 \$0 1 \$3000 0 \$0 .2 \$4000 .8 \$0 .25 \$3000 .75 \$0 Suppose you prefer A1 to B1, but B2 to A2 Are your preferences summarized by a von- Neumann/Morgenstern utility function? Let u0, u1, and u2 be your Bernoulli payoffs to \$0, \$3000, and \$4000 respectively Your Choice of Gambles A1= B1= A2= B2= .8 \$4000 .2 \$0 1 \$3000 0 \$0 .2 \$4000 .8 \$0 .25 \$3000 .75 \$0 A1 better than B1 implies (.2)u2+(.8)u0 > (.25)u1+(.75)u0 (.2)u2+(.05)u0 > (.25)u1 (.8)u2+(.2)u0 > u1 B2 better than A2 implies (.8)u2+(.2)u0 < (1)u1 OR OR Your Choice of Gambles The fact that a lot of people prefer A1 to B1, but prefer B2 to A2 is known as the Allais Paradox, named after its inventor, Maurice Allais Allais won the Nobel Prize in Economics in 1988 Allais has also “discovered” a certain anisotropy of space in contradiction to Einstein’s Theory of Relativity Characterization of M-S NE We can characterize mixed-strategy NE with two properties: for a given m-s NE, and for each player Every pure strategy receiving positive weight in the player’s mixed strategy yields the same expected payoff (“opponent’s indifference property”) Any pure strategy receiving zero weight yields an expected payoff no higher than the ones

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