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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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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

80

100

Quantity


Games and best responses

  • What is Bob’s optimal quantity as a monopolist?

  • It turns out to be 40 units

  • Suppose now Ann is contemplating entry

  • How will the total quantity be allocated between the two?


Games and best responses

  • It depends on how they expect each other to act

  • Suppose Ann is a sophisticated thinker and responds optimally to Bob

  • Ann’s best response to Bob is (80-QB)/2

  • What should Bob do?

  • Note:

    Bob wants the total quantity to be the midpoint between Ann’s quantity (QA) and the break-even point (80)


Games and best responses

  • Demand curve facing Bob is

  • To maximize profits, 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


Games and best responses

  • Four scenarios for Bob’s reasoning


What if Bob’s quantity is a given?

  • How many bunches should Bob bring, knowing how Ann would react to him?

  • Now, total quantity is (80-QB)/2+QB= (80+QB)/2

  • Thus, mkt price is $10-$0.10{(80+QB)/2}

  • Bob’s optimal quantity is 40; and so Ann’s quantity is 20

  • A significant first-mover advantage!

  • Why does Bob produce monopoly quantity?


Strategic-Form Games

  • Elements of strategic-form games:

    - 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)

  • Each player goal is to maximize his own payoff, and NOT to ‘beat’ other players

  • A Zero-Sum game is where =0

  • Most applications in the Social Sciences are non zero-sum games


Mixed-Strategies

  • A mixed-strategy σiis a probability distribution over pure-strategies

  • σi(si) is probability that σi assigns to si

  • Space of player i’s mixed-strategies is ∑i

  • Space of mixed-strategy profiles is ∑=with elements σ

  • Each player’s randomization is independent of those of other players

  • Player i’s payoff to profile σ is


Computing payoff from a mixed-strategy

  • Player 1 plays along rows; 2 along columns

  • Let σ1={⅓, ⅓, ⅓} andσ2={0, ½, ½ }.

  • What is player 1’s payoff?


Dominated Strategies

  • Is there an obvious prediction about how the previous game will be played?

  • Iterated Strict Dominance predicts (U, L)

  • Let “–i” denote all players other than i

  • Then strategies of other players s-i S-i and a strategy profile is (si , s-i )

  • Defn: Pure strategy si is strictly dominated for player i if there exists ∑i such that,

    for all s-i S-i


Some implications of dominance

  • A mixed-strategy that assigns positive probability to a dominated pure strategy, is dominated

  • A mixed-strategy my be dominated even though it assigns positive probability only to undominated pure strategies.

  • Examples: -Prisoner’s Dilemma

    -Second-Price Auction


Nash Equilibrium

  • Defn: A mixed-strategy profile σ* is a Nash Equilibrium if, for all players i,

    for all for all s-i S-i

  • Common examples in economics:

    - Cournot quantity-setting game

    - Bertrand price-setting game


Assumptions

  • Rationality

    • Players aim to maximize their payoffs

    • Players are perfect calculators

  • Common Knowledge

    • Each player knows the rules of the game

    • Each player knows that each player knows the rules

    • Each player knows that each player knows that each player knows the rules

    • Each player knows that each player knows that each player knows that each player knows the rules

    • Each player knows that each player knows that each player knows that each player knows that each player knows the rules

    • Etc. Etc. Etc.


  • Properties and implications

    • Property 1: A player must be indifferent between all pure strategies in the support of a mixed-strategy

    • Property 2: One only needs to check for pure-strategy deviations

    • Property 3: If a single strategy survives iterated deletion of strictly dominated strategies it is a Nash Equilibrium

    • Question: Why play mixed strategies when all pure strategies in the support have same payoff?


    Further comments on N.E.

    • In many games pure strategy NE do not exist

      -example is game of “Matching Pennies”

    • Sometimes there are multiple Nash Equilibria

      - “Battle of the Sexes”; “Chicken”

    • Schelling’s theory of “focal points”

    • Pre-play communication


    Baseball anyone?

    • 1986 baseball National League championship series

    • The New York Mets won a crucial game against he Houston Astros

    • Len Dykstra hit Dave Smith’s second pitch for a two-run home run

    • Later the two players talked about this critical play


    Analysis of a home run

    • Dykstra said, “He threw me a fastball on the first pitch and I fouled it off. I had a gut feeling then that he’d throw me a forkball next, and he did. I got a pitch I saw real well, and I hit it real well”.

    • Smith said, “What it boils down to is that, it was a bad pitch selection…if I had to do it over again, it would be [another] fastball”.

    • Would Dykstra not have been prepared for a fastball?

    • Again, randomization is the only way to go


    But how do you randomize?

    • In a game of tennis, suppose receiver’s forehand is stronger than backhand

    • Consider following probabilities of successfully returning serve

    Server’s Aim

    Receiver’s Move


    Reducing receiver’s effectiveness by randomizing

    • Suppose server tosses a coin before each serve

    • Aims to forehand or backhand according to coin turning heads/tails

    • When receiver moves to forehand, his successful return rate is 55%

    • When receiver moves to backhand, his successful return rate is 45%

    • Given server’s randomization, receiver should move to forehand

    • The server has already an improved outcome compared to serving the same way all the time!!


    What is server’s best mix?

    • Consider following graph

    90

    Anticipate Forehand

    60

    Anticipate Backhand

    48

    Percentage successful returns

    30

    20

    0

    40

    100

    Percentage of times server aims serve to forehand


    The mixing probabilities

    • The 40:60 mixture of forehands to backhands is the equilibrium

    • This mixture is the only one that cannot be exploited by the receiver to his own advantage

    • With this mixture the receiver does equally well with either of his choices

    • Both ensure the receiver a success rate of 48%


    Nash bargaining

    • Two players, N=1, 2, bargain over the partition of a cake

    • The set of Agreements, A, has elements

      A={(a1, a2)єR2: ai≥0 for i=1, 2}

    • The event of Disagreement is D

    • Players have utilities ui, i=1, 2, s.t. ui: A {D}→R

    • Let, S={(u1(a), u2(a)) for a є A}, and d=(u1(D), u2(D))

    • A Bargaining ProblemB is the set of pairs <S, d>, and the Bargaining Solution is a function f :B → R2 that assigns to each <S, d> єB an element of S


    Nash’s axioms

    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).


    Nash bargaining solution

    Theorem (Nash 1950)

    There is a unique bargaining solution fN:B → R2 satisfying the above axioms given by

    fN(S, d)=


    Application: Splitting a dollar

    • Set of agreements is

      A={(a1, a2)єR2: a1+ a2≤1 and ai≥0 for i=1, 2}

    • Players are risk-averse, i.e. uiare concave

    • Disagreement point is d=(u1(0), u2(0))=(0, 0)

    • So <S, d> is a bargaining problem

      Result 1:With equal risk-aversion, players are symmetric & SYM, PAR give (u(1/2), u(1/2))


    Role of risk-aversion

    • Let player 2 be more risk-averse than player 1

    • Let player 2’s utility be v2 =hu2,where h is concave, and v1 = u1

    • Let <S/, d/> be bargaining problem with utilities vi

    • The optimizing program for <S, d> gives solution zu where zu solves:

    • The optimizing program for <S/, d/> gives solution zv where zv solves:


    Role of risk-aversion

    • The first program gives,

    • The second program gives,

    • Result 2:If player 2 becomes more risk-averse, then Player 1’s share of the dollar in the Nash solution increases.


    Existence of NE

    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


    Existence of NE (cont)

    Theorem (Shizuo Kakutani 1941)

    If :

    • ∑is compact and convex

    • A correspondence r(σ) :∑→∑ is non-empty and convex

    • r(σ) has a closed graph

      Then r(σ) has a fixed point, that is, there exists σ*

      such that, r(σ*)= σ*


    Nash equilibrium is a fixed point

    • For any strategy profile σ, player i’s reaction correspondence ri maps σ to the mixed strategy that maximizes his payoff, given his opponents play σ-i . Thus, ri (σ)= ri (σi ,σ-i ) = ui(σ/i , σ-i )

    • Each player i=1,2,..n, has a reaction function

    • Let us form the n-tuple, r(σ)=(r1 (σ), r2 (σ),…, rn(σ)), where r(σ)є∑

    • Suppose there exists σ* є∑such that,for all players, σ*i =ri(σ*)= ri(σ*i , σ*-i), then σ* is a Nash equilibrium

    • But σ*i =ri(σ*) for all i,means that σ*=r(σ*), i.e. σ* is a fixed point of the reaction correspondence r(.)


    Applying Kakutani

    • ∑ is compact and convex:

      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

    • r(σ) is non-empty:

      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


    Applying Kakutani

    • r(σ) is convex:

      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)

    • r(σ) has a closed graph:

      If with , then


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