Lecture III: Normal Form Games

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Lecture III Normal Form Games

• Recommended Reading
• Dixit Skeath Chapters 4, 5, 7, 8
• Gibbons Chapter 1
• Osborne Chapters 2-4

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Recap Introduction

• A Game
• Players Strategy Set (rules plans of action)
  Outcomes (payoffs)
• Nash Equilibrium
• A best response to a best response
• i.e., no player wants to alter strategy
  unilaterally
• If G S1,, Sn u1,, un, the strategies
  (s1,,sn) are a Nash equilibrium if ? i
  ui(s1,,si-1, si, si1,, sn)
  ui(s1,,si-1, si, si1,, sn)

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Prisoners Dilemma

• Normal form representation
• Players have discrete strategies, confess, stay
  silent
• Cells contain payoffs, rows first, columns
  second

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Prisoners Dilemma

• Identify NE by eliminating strictly dominated
  strategies
• For i, a strategy, si, is strictly dominated by
  si if

• ui(s1...si-1, si, si1...sn) lt ui(s1...si-1,
  si, si1...sn)
• ? (s1...si-1, si, si1...sn) ? S
• i.e., i always does better not playing si
  irrespective of other players strategies

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Prisoners Dilemma

• Conversely, a strictly dominant strategy
  maximizes ui irrespective of what others do
• i should never play a strictly dominated
  strategy
• If a strictly dominant strategy exists, i should
  play it.
• But NE does not hinge on existence of strictly
  dominant strategies

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Prisoners Dilemma

• Player is Logic
• Take js strategy as fixed
• Compare payoffs under different strategies
• Given sj Silent
• ui(si(C), sj(S)) 4
• ui(si(S), sj(S)) 3
• Given sj Confess
• ui(si(C), sj(C)) 2
• ui(si(S), sj(C)) 0

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Prisoners Dilemma

• Player is Logic
• Take js strategy as fixed
• Compare payoffs under different strategies
• Given sj Silent
• ui(si(C), sj(S)) 4
• ui(si(S), sj(S)) 3
• Given sj Confess
• ui(si(C), sj(C)) 2
• ui(si(S), sj(C)) 0

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Prisoners Dilemma

• Player is Logic
• Take js strategy as fixed
• Compare payoffs under different strategies
• Given sj Silent
• ui(si(C), sj(S)) 4
• ui(si(S), sj(S)) 3
• Given sj Confess
• ui(si(C), sj(C)) 2
• ui(si(S), sj(C)) 0

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Weak Dominance

• Some games do not have strictly dominant
  strategies
• Battle of Bismark Sea
• US has no dominant strategy
• For Japan, N weakly dominates S, (i.e., N at
  least as good as S no matter what US does, and
  sometimes better than S)
• This allows US to choose strategy generates NE

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Weak Dominance

• Elimination of weakly dominated strategies not
  sufficient to identify all NE.
• US Canada run on 110V (convenient)
• Both switching to 220V brings world convergence
  extra convenience
• If only one switches, world convenience offset by
  continental inconvenience
• For US Canada U(220V) ? U(110V) irrespective
  of others strategy
• 220V, 220V is NE...
• 110V, 110V also NE If US is 110V would Canada
  unilaterally switch to 220V?

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Multiple Equilibria

• Not all games have unique NE
• Consider the following games
• Pure Coordination
• Battle of the Sexes
• Stag Hunt

Carla
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Multiple Equilibria

• Not all games have unique NE
• Consider the following games
• Pure Coordination
• no reason to for i to resist L, L over R, R
• but no compelling reason for i to play L or R

Carla
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Multiple Equilibria

• Not all games have unique NE
• Consider the following games
• Battle of the Sexes
• conflict coordination
• still no reason to choose C, C over T, T
• Credible commitment?

Carla
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Multiple Equilibria

• Not all games have unique NE
• Consider the following games
• Stag Hunt
• Pareto optimality provides compelling reason for
  F, F
• but P, P remains NE despite P being weakly
  dominated by F

Carla
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No Pure Strategy Nash Equilibria

• Some games do not have pure strategy NE
• Consider reformulation of Battle of Bismark Sea
  game
• Neither player has a dominant strategy
• At every cell, at least one player want to alter
  strategy
• No NE!

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Mixed Strategies

• If i has k 1K pure strategies, Si
  si1,,siK in G S1,, Sn u1,, un, then a
  mixed strategy is a probability distribution, pi
  (pi1,,piK) s.t. 0 pik 1 and ?pik 1
• If G S1,,Sn u1,,un, where n is finite and
  Si is finite ? i ? ? at least one NE, possibly
  involving mixed strategies
• Informally, i plays all her available pure
  strategies with some probability (perhaps Pr 0
  for some)
• Interpret is mixed stratgy as js uncertainty
  about what strategy i will actually adopt

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Mixed Strategies in Practice

• US plays N with Pr p Japan plays N with Pr q
• US
• EU(N) 4-2q
• EU(S) 1 4q
• EU(N) gt EU(S) iff ½ gt q

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Mixed Strategies in Practice

• US plays N with Pr p Japan plays N with Pr q
• Japan
• EU(N) 2p
• EU(S) 4 3p
• EU(N) gt EU(S) iff p gt 4/5

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Best-Response Curves
1

• If q lt 1/2, US should play N with Pr(p) 1
  (it gets more utility)
• If p gt4/5, Japan should play N with Pr(q) 1
  (ditto)

q
1/2
0
1
4/5
p
USs best-response curve Japans best-response
curve
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Continuous Strategies The Cournot Game

• Two firms in competition, i j
• qi and qj denote quantities of single, homogenous
  good produced by each firm
• P(Q) a Q is market clearing price, where Q
  qi qj.
• N.B. (If a lt Q, P(Q) 0, but lets assume a gt
  Q.)
• Ci(qi) cqi, i.e., constant costs per unit

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Continuous Strategies The Cournot Game

• Each firms strategy space is Si 0, ?) so any
  qi 0 is admissible (though a puts an implicit
  limit on qi).
• Each firms payoffs are equal to their revenues,
  i.e., market price ? quantity costs
• ?i(qi, qj) qiP(qi qj) c qia (qi
  qj) c
• Firms choose quantities simultaneously how much
  does / should each produce?

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Continuous Strategies The Cournot Game

• Each firms faces an optimization problem
• max ?i(qi, qj) max qia (qi qj) c
  
• 0 q ? 0 q ?
• To solve, we need to obtain is first-order
  condition, i.e., differentiating above w.r.t qi,
  set equal to 0, and solve
• qi ½(a qj c)
• Game is symmetric in strategies payoffs, so
• qj ½(a qi c)

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Continuous Strategies The Cournot Game

• Each firms wants to produce
• qi ½(a qj c) 1
• qj ½(a qi c) 2
• Equations 1 2 tells us what qi and qj are, so
  substitute qj from 2 into Equation 1 and solve
• qi ½(a ½(a qi c) c)
• (a c)/3
• Same holds for j. The NE qi qj (a c)/3

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Best-Response Curves

• Drawing best response curves helps to get a
  better sense of the NE
• The NE occurs at the intersection of each firms
  payoff (i.e., revenue) curve.

qj
qia-(qiqj)-c
(a c)/3
0
(a c)/3
qi
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Combining Discrete Continuous Strategies

• Lichbach (1990) provides examples of games in
  which players strategies are discrete but their
  payoffs are continuous
• Just replace the 1s, 2s 3s etc in our earlier
  examples with a payoff function defined by
  variables as in the Cournot game
• e.g., if for Row B gt D gt A gt C and for Column C gt
  D gt A gt D, then the game has the form of a
  Prisoners Dilemma

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Combining Discrete Continuous Strategies

• In contrast, if for Row A D gt B C, for
  Column, A D gt B C, then the game is a pure
  coordination game
• We can replace A, B, C, D with functions if we
  wished.
• Proving that I, I is a unique NE then requires
  showing conditions under which
• uR(sR(I), sC(.)) gt uR(sR(II), sC(.))
• uR(sR(I), sC(.)) gt uR(sR(II), sC(.))
• i.e., that for Row Column, II is strictly
  dominated by I