Strategy14Handout - 10/23/2008 Your Choice of Gambles A1=...

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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 .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 .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 .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
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This note was uploaded on 04/14/2010 for the course ECON 398 taught by Professor Emre during the Spring '07 term at University of Michigan.

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Strategy14Handout - 10/23/2008 Your Choice of Gambles A1=...

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