Strategies 3/3/2009
Mixed Matching Pennies
We denote the strategies on the game bi-matrix
Guildenstern q Heads Rosencrantz R t p 1-p Heads Hd Tails 1 , -1 1 -1 , 1 1-q Tails -1 , 1 1 1 , -1
Preferences Involving Gambles
We revisit our description of payoffs in the game bimatrix. Consider Tom, who is faced with two outcomes
A: get $1 B: get $4
Guildensterns payoff to H: (-1)p + 1(1-p) = 1-2p T: (1)p + (-1)(1-p) = 2p-1
Tom likes money, the more the better. So Tom thinks that outcome B is better than A
Preferences Involving Gambles
Toms preferences are summarized by
u1(m)=m, so that u1(1)=1, and u1(4)=4 u2(m)=m, so that u2(1)=1, and u2(4)=16
u3(m)=m, so that u3(1)=1, and u3(4)=2 2
Preferences Involving Gambles
We would like to consider preferences over gambles (lotteries) involving outcomes Can we come up with a utility function that ranks preferences over g p gambles ( (involving g outcomes) along with outcomes themselves? Example: Suppose G is the gamble
Get $1 with probability , and Get $4 with probability
any of one these payoff (utility) functions so long as outcomes are get some money
Preferences Involving Gambles
Note that the expected value of G, E[G] = ($1) + ($4) = $2.50 Must Tom think that G is just as good as $2.50? No Maybe, Tom thinks G is better than $2.50, or maybe Tom thinks G is not as good as $2.50
Preferences Involving Gambles
Suppose we have a utility function over outcomes u:{outcomes} - the real numbers that represents a particular set of outcomes It would be nice to have a utility function v:{gambles over outcomes} that ranks gambles over outcomes as well as outcomes
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Preferences Involving Gambles
Desirable Property #1: Since outcomes themselves can be thought of as gambles, v should agree with u on such degenerate gambles. For example, let F be the gamble
Get $4 with probability 1
Preferences Involving Gambles
Desirable Property #2 the expected utility property Let H be the gamble
Get $m1 with p $ probability p y p1 Get $m2 with probablity p2 yada yada yada Get $mk with probability pk
Note that pk = 1 - p1 - p2 - pk-1 Should have v(F)=u($4)
Preferences Involving Gambles
Preferences that are represented by the utility function u:{outcomes} are said to have the expected utility property if preferences over gambles can be represented by a utility function v such that v(H) = p1u(m1) + p2u(m2) + + pku(mk)
Preferences Involving Gambles
If a persons preferences over gambles are represented by a utility function with the expected utility property, then the preferences are said to be von Neumann-Morgenstern preferences A payoff/utility function with the expected utility property is called a Bernoulli payoff/utility function. In this case, preferences over gambles are summarized by expected utility.
Preferences Involving Gambles
Read: Dixit & Skeath, Chpt 7 Appendix pt. 2 It is not always the case that a persons preferences over gambles can be summarized by expected utility. yp y In this class, unless otherwise stated, we assume that payoffs in bi-matrices are Bernoulli payoffs named after Daniel Bernoulli (1700-1782)
St. Petersburg Paradox
Expected utility was invented by Daniel Bernoulli to solve the St. Petersburg Paradox The St. Petersburg Paradox was invented by Daniels cousin Nicolas Bernoulli in 1713 to give g his family a hard time Let the St. Petersburg gamble be
Get $1 with probability Get $2 with probability Get $4 with probablity yada yada yada yada
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St. Petersburg Paradox
How much would you pay to be the beneficiary this of gamble? Note that the expected value of this gamble is ($1) + ($2) + ($4) + = $.50 + $.50 + $.50 + = $ Why are you only willing to pay a few bucks for this fantastically valuable gamble? Daniel Bernoulli proposed a solution: you are risk aversethe high payoffs are unlikely enough that you are scared to lose money, and so only willing to pay a little for this gamble
Preferences Involving Gambles
Recall Tom and the outcomes A: get $1 and B: get $4, and the Gamble G get $1 with probablity , get $4 with probablity We saw that E[G] $2.50 E[G]=$2.50 Suppose the utility functions u1(m)=m, 2 u2(m)=m, and u3(m)=m are Bernoulli payoff functions. Do they summarize the same preferences over gambles?
Preferences Involving Gambles
By the expected utility property u1(G) = u1($1) + u1($4) = 1 + 4 = 2.50 2 50 = u1($2.50) therefore G is just as good as $2.50 according the u1 preferences
Preferences Involving Gambles
By the expected utility property u2(G) = u2($1) + u2($4) = 1 + 16 = 8.50 8 50 > 6.25 = u2($2.50) therefore G is better than $2.50 according the u2 preferences
Preferences Involving Gambles
By the expected utility property u3(G) = u3($1) + u3($4) = 1 + 2 = 1.50 1 50 < (2.50) ~1.58 = u3($2.50) therefore G is not as good as $2.50 according the u3 preferences
Preferences Involving Gambles
The preferences u1, u2 and u3 represent different preferences over gambles because they rank G and $2.50 differently
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Preferences Involving Gambles
18 16 14 12 10 8 6 4 2 0 0 1 2 3 4 money 5 u2($2.50)
Preferences Involving Gambles
18 16 14 12 10 8 6 4 2 0 0 1 2 3 4 money 5 u2($2.50) u2(G) = u2($1)+u2($4)
u2
u2
Preferences Involving Gambles
The preferences u1, u2 and u3 represent different preferences over gambles because they rank G and $2.50 differently.
u2 represents risk-loving preferences
Preferences Involving Gambles
2.5 2 1.5 u3($2.50) 1 0.5 0 0 1 2 3 4 money 5
u3
Preferences Involving Gambles
2.5 2 1.5 u3($2.50) 1 u3($1)+u3($4) 0.5 0 0 1 2 3 4 money 5 =u3(G)
Preferences Involving Gambles
The preferences u1, u2 and u3 represent different preferences over gambles because they rank G and $2.50 differently.
u2 represents risk-loving preferences u3 represents risk-averse preferences
u3
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Preferences Involving Gambles
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 money 5 u1($1)+u1($4) =u1(G) u1($2.50)
Preferences Involving Gambles
The preferences u1, u2 and u3 represent different preferences over gambles because they rank G and $2.50 differently.
u2 represents risk-loving preferences u3 represents risk-averse preferences u1 represents risk-neutral preferences
u1
Preferences Involving Gambles
Expected utility was first proposed by Daniel Bernoulli in 1738 to solve the St. Petersburg Paradox You are risk averse because you dont value y large payoffs much more than smaller ones Therefore the expected utility of the St. Petersburg gamble is small The theory of expected utility was developed by von Neumann and Morgenstern in Theory of Games and Economic Behavior (1944)
Preferences Involving Gambles
Consider these games with Bernoulli payoffs
L R 16 , 4 9,9
Game A
U D
1 , 16 4,1
L
R 256 , 16 81 , 81
Game B
U D
1 , 256 16 , 1
L
R 4,2 3,3
Game C
U D
1,4 2,1
Preferences Involving Gambles
If we do not allow for randomization, all these games are strategically equivalent If we allow for randomization, all these games are different, since they express different yp attitudes toward randomness. From now on, we assume payoffs are Bernoulli payoffs
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