Static Games of Complete Information. Games and best responses. Bob is a florist One bunch of flowers sells for $10, but to sell 100 bunches market price should be zero Thus, each extra bunch reduces market price by a dime Cost is $2 a bunch. Games and best responses. $ 10. Price. $ 2.
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Static Games of Complete Information
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$ 10
Price
$ 2
80
100
Quantity
Bob wants the total quantity to be the midpoint between Ann’s quantity (QA) and the break-even point (80)
$ 10
Price
$ 2
QA
(80+QA)/2
80
100
Quantity
Bob
- finite set of players, i I ={1,2,…,I}
- pure-strategy space Si for each player i
- payoff functions ui(s) for each player i and for each strategy profile s=(s1, s2,…, sI)
for all s-i S-i
-Second-Price Auction
for all for all s-i S-i
- Cournot quantity-setting game
- Bertrand price-setting game
-example is game of “Matching Pennies”
- “Battle of the Sexes”; “Chicken”
Server’s Aim
Receiver’s Move
90
Anticipate Forehand
60
Anticipate Backhand
48
Percentage successful returns
30
20
0
40
100
Percentage of times server aims serve to forehand
A={(a1, a2)єR2: ai≥0 for i=1, 2}
1. INV (invariance to equiv utility representations):
If <S/, d/> is obtained from <S, d> by si ,
then, fi(S/, d/)=
2. SYM (symmetry)
If <S, d> is symmetric, then f1(S, d)= f2(S, d)
3. PAR (Pareto efficiency)
Suppose <S, d> is a bargaining problem, with si , ti єS , and ti >si , for i=1,2, then f(S, d)≠s
4. IIA (independence of irrelevant alternatives)
If <S, d> and <T, d> are bargaining problems with S T and f(T, d) єS, then f(S, d) = f(T, d).
Theorem (Nash 1950)
There is a unique bargaining solution fN:B → R2 satisfying the above axioms given by
fN(S, d)=
A={(a1, a2)єR2: a1+ a2≤1 and ai≥0 for i=1, 2}
Result 1:With equal risk-aversion, players are symmetric & SYM, PAR give (u(1/2), u(1/2))
Theorem (Nash 1950)
Every finite strategic-form game has a mixed-strategy equilibrium.
Sketch of Proof:
1. Use players’ Reaction Correspondences r(σ)
2. Realize that a NE is a Fixed Point of r(σ)
3. Show that conditions for Kakutani’s Fixed Point Theorem are satisfied in this case
Theorem (Shizuo Kakutani 1941)
If :
Then r(σ) has a fixed point, that is, there exists σ*
such that, r(σ*)= σ*
1. Mixed strategies convexify the strategy space, and so each ∑i is a convex space of dimension (#Si-1)
2.Theorem (Heine-Borel):
Closed & bounded subsets in Rn are compact
1.Theorem (Weierstrass):
A continuous real-valued function defined on a compact space achieves its maximum values
2.Player’s i’sutility functions are continuous in σi
Suppose σ/,σ//є r(σ) . uiis linear, therefore for λє(0, 1)
ui(λσi/+ (1- λ)σi//, σ-i)= λ ui(σi/, σ-i) + (1- λ)ui(σi//, σ-i). Thus, if σi/,σi//are best responsestoσ-i, then so is their weighted average. Thus, λσi/+ (1- λ)σi// є r(σ)
If with , then