Applied Microeconomics

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Applied Microeconomics

• Game Theory I Strategic-Form Games

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Outline

• Strategic-form games
• Dominance and dominance solvable games
• Common knowledge
• Nash equilibrium
• Mixed strategies
• Mixed Nash Equilibrium

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Readings

• Kreps Chapter 21
• Perloff Chapter 13
• Zandt Chapter 9

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Introduction

• So far we assumed that the firm maximizes profit
  assuming its competitors remain passive
• This is a reasonable assumption if the firm is
  one of many small firms as in a competitive
  market or if the firm is a monopoly producing of
  a good with no close substitutes

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Introduction

• However, in an oligopoly, a market with a few
  firms that have some market power, we need to
  relax this assumption and allow for strategic
  interaction
• To model this situation we need a new tool called
  non-cooperative game theory

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Example

• Alpha has a local monopoly in the market for a
  good and is making a profit of 2 million Euros
• Beta is considering entering the market
• If Beta enters the market, Alpha could either
  fight by setting a low price or accommodate, by
  setting a high price
• If Alpha fights, then both firms get a profit of
  1 million Euros
• If Alpha accommodates, then both firms get a
  profit of 1 million Euros

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Example

• We could model this strategic interaction in the
  following ways

Extensive-form game
Strategic-form game
Beta
Alpha
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Game Theory

• Game theory
• Cooperative
• Non-cooperative
• Non-cooperative game theory
• Sequential/extensive-form games
• Strategic-form/normal-form games
• Strategic-form games
• Infinite action
• Finite action

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Strategic-Form Games

• Static analysis
• An extensive-form game can be represented in
  strategic form
• Infinite action or finite action games
• Infinite and finite games

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Examples of Finite Action Games
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Example of an Infinite Action Game

• Three firms, Alpha, Beta, and Gamma,
  simultaneously and independently decides an
  amount xi to spend on advertisement, where xi is
  non-negative
• Total sales in the market is 1 and each firm
  sells in proportion to its share of total ad
  expenditure
• Alphas profit is xAlpha/(xAlpha xBeta
  xGamma)-xAlpha and Betas is xBeta/(xAlpha
  xBeta xGamma)-xBeta, and Gammas is
  xGamma/(xAlpha xBeta xGamma)-xGamma

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Concepts

• A strategy si is a complete contingent plan for a
  player in the game
• Example 1 sifink
• Example 2 si2/9
• A strategy profile s is a vector of strategies,
  one for each player
• Example 1 s(fink,fink)
• Example 2 s(xAlpha2/9, xBeta2/9, xGamma2/9)

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Concepts

• A payoff function ui(si,s-i) gives the payoff or
  utility to player i for any of her strategies si
  and any of her opponents strategies s-i
• Example 1 U1(fink,fink)0, U1(fink,cooperate)2,
  U1(cooperate,fink)-1,U1(cooperate,cooperate)1
• Example 2 UAlpha(xAlpha, xBeta, xGamma)
  xAlpha/(xAlpha xBeta xGamma)-xAlpha
• Payoffs are ordinal (we could f.i. multiply all
  of them by a positive constant without changing
  the game), and the players are generally assumed
  to be expected utility maximizers

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Strategic Form Games

• A strategic-form game is
• A set of players N1,2,,n
• A set of strategies for each player i
  Sisi1,,siJ
• A payoff function for each player i ui(si,s-i)

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Predicting Play Dominance

• A strategy si is dominated for player i if there
  is some other strategy si that gives a weakly
  better payoff no matter what strategies the other
  players use, and strictly better payoff for some
  s-i si is dominated if there is si? si such
  that ui(si,s-i)ui(si,s-i) for any s-i and
  ui(si,s-i)ltui(si,s-i) for some s-i
• A strategy si is strictly dominated for player i
  if there is some other strategy si that gives a
  strictly better payoff no matter what strategies
  the other players use si is strictly dominated
  if there is si? si such that ui(si,s-i)ltui(si,s-
  i) for any s-i

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Example

• If we assume that no player would ever play a
  strictly dominated strategy, we can sometimes
  make predict the strategies that will be played
• Is any strategy strictly dominated in any of
  these games?

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Iterated Elimination of Dominated of Strategies

• Eliminating strictly dominated strategy rarely
  gives precise predictions, but the procedure can
  be extended
• Suppose we delete all the strictly dominated
  strategies from the strategy set S1 of the game
  G1
• Once we have done this we get a new game G2 with
  strategy set S2 from which we once again can
  delete the strictly dominated strategies
• We can keep on deleting strategies in this way
  until we reach a game Gt with strategy set St
  such that no more strategies can be deleted

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Example

• Find the set of strategies that survives iterated
  elimination of strictly dominated strategies in
  the following game

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Iterated Elimination of Dominated of Strategies

• This procedure is called iterated elimination of
  strictly dominated strategies
• In finite games (and most infinite games), the
  set of strategies that survives iterated
  elimination of strictly dominated strategies does
  not depend on the order in which we eliminate
  strategies!
• If only one strategy for each player survives the
  procedure, then the game is said to be dominance
  solvable

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What Assumptions Would Result in IESDS?

• Deduction If all players know their payoff
  function and never play a strictly dominated
  strategy, know that their opponent knows his own
  payoff function and never plays a strictly
  dominated strategy, know that their opponent
  knows that they know their payoff function and
  never play a strictly dominated strategy etc.
• Evolutionary selection Alternatively, we can
  assume that players recurrently are drawn from a
  large population of individuals to play the game,
  and that individuals that perform worse disappear
  or change strategy

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Experimental Evidence

• Few subjects play dominated strategies
• More subjects play iteratively dominated
  strategies
• The more iterations are needed to solve game, the
  worse predictive power of IESDS

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Iterated Elimination of Weakly Dominated
Strategies

• The set of strategies that survive iterated
  elimination of weakly dominated strategies in
  finite action games may depend on the order of
  elimination (compare eliminating b and B)

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Nash Equilibrium

• Often many strategies iterative elimination of
  strictly dominated strategies
• We want to be able to make predictions also in
  this case
• The most famous concept for doing this is the
  Nash Equilibrium

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Nash Equilibrium

• A Nash equilibrium is a strategy profile such
  that no player could gain by deviating
  unilaterally and play a different strategy
• Formally, s is a NE if for all i in N, and all
  si in Si, ui(si,s-i)ui(si,s-i)
• If the inequality is strict, then s is said to
  be a strict NE

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Examples
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Finding Nash Equilibria

• In order to find the Nash equilibria of a game it
  is useful to find each players best reply to any
  strategy played by the opponent(s) the best
  reply correspondence
• A NE is a strategyprofile such that allplayers
  are playinga best reply

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What Assumptions Would Result in NE?

• Deduction
• Evolutionary selection
• Learning

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Nash Equilibrium and Iterated Elimination

• A strategy that is eliminated by iterated
  elimination of strictly dominated strategies
  cannot be part of a NE
• If the game is dominance solvable, the surviving
  strategy profile is the unique NE of the game
• If the game is dominance solvable using iterated
  elimination of weakly dominated strategies, the
  resulting strategy profile is a NE, but there may
  be more NE

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Example Without Nash Equilibrium in Pure
Strategies

• Sometimes, there does not exist a NE is pure
  strategies
• To deal with this we define mixed strategies pi
  as probability distributions over the set of pure
  strategies in Si
• Motivation
• Each player is randomizing
• Game played against a randomly drawn individual
  from a large population where a share pi1 play
  strategy si1, a share pi2 play strategy si2 etc.

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Example

• Suppose two players are playing the following
  game
• A mixed strategyfor player onecould be
  p1(0.3,0.7) to play A with prob. 0.3 and B
  with prob. 0.7

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Nash Equilibrium in Mixed Strategies

• Let ?Si set of all mixed strategies over Si, and
  let p be a mixed strategy profile, p(p1,p2,,p3)
• Formally, p is a NE if for all i in N, and all
  pi in ?Si, ui(pi,p-i)ui(pi,p-i)
• Nash (1950) Any finite game has at least one
  Nash equilibrium in mixed strategies
• Not that pure strategies are just mixed
  strategies that play a particular pure strategy
  with probability one

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Example

• Find a NE in the above game
• If there is a NE such that player 1 is
  randomizing when 2 is playing a with prob. p and
  b with prob. 1-p, then player 1 must get the same
  expected payoff from both of his strategies
• U(A)-2p1(1-p)1-3p
• U(B)1p-1(1-p) 2p-1
• Hence, 1-3p2p-1 or p2/50.4

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Example

• Likewise, if there is a NE such that player 2 is
  randomizing when player 1 is playing A with prob.
  q and B with prob.1-q, then player 2 must get the
  same expected payoff from both of his strategies
• U(a)1q-1(1-q)2q-1
• U(b)-1q1(1-q)1-2q
• Hence, 2q-11-2q or q2/40.5
• The NE is thus given by ((qA0.5,qB0.5),((pA0.4,
  pB0.6))

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The Best-Reply Correspondence

• Let G be a strategic-form game with players
  i1,,n, strategy sets S1,,Sn, and payoff
  functions u1,,un
• For each player, define player is best-reply
  correspondence Bi(s-i) that maps any strategy
  choice s-i by the opponent(s) to the most
  profitable strategy for player i
• Mathematically Bi(s-i)siui(si,s-i)ui(si,s-i
  ) for all si in Si

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Nash Equilibrium and the Best Reply Correspondence

• We can define a NE using the best-reply
  correspondence it is a strategy profile s such
  that all players are playing a best reply
• Mathematically s is an NE if si?Bi(s-i) for
  i1,,n
• Hence, the NE can be found using the best-reply
  correspondences of all players

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Example Finite Game

• Suppose the column player plays a with prob. Pa
• The row players expected utility of A is then
  U(A)2Pa-5(1- Pa)-57Pa
• The row players expected utility of B is then
  U(B)0Pa0(1- Pa)0
• Hence, A is a best reply for Pa5/7 and B for
  Pa5/7 and any randomization over the two for
  Pa5/7

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Example Finite Game
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Example Finite Game

• Suppose the row player plays A with prob. PA
• The column players expected utility of a is then
  U(a)3PA-5(1- Pa)-58PA
• The row players expected utility of b is then
  U(b)0PA0(1- PA)0
• Hence, a is a best reply for PA5/8 and B for
  PA5/8 and any randomization over the two for
  PA5/8

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Example Finite Game
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Example Finite Game

• The NE are given by the points where B1(Pa)PA
  and B2(PA)Pa
• Graphically this is where the two best-reply
  correspondences intersect
• Hence, the NE of the game are (PA0,Pa0),
  (PA1,Pa1), and (PA5/8,Pa5/7)

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Example Quantity Competition

• Suppose two firms, iAlpha, Beta, are competing
  in quantities and simultaneously and
  independently decide a non-negative amount xi to
  produce (Cournot competition)
• Each firm has per unit cost clt1 and the market
  inverse demand function is given by
  P(xi,x-i)1-(xix-i) for xix-ilt1 and 0 otherwise
• Payoffs are given by uAlpha(xAlpha,xBeta)
  xAlpha(1-(xAlphaxBeta)-c)uBeta(xBeta,xAlpha)
  xBeta(1-(xAlphaxBeta)-c)

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Example Quantity Competition

• To find the best-reply correspondence for firm i
  we solve Maxxi0 xi(1-(xix-i)-c)with the
  first-order condition 1-(x-ixi)-c-xi0
• Solving for xi gives Bi(x-i)(1-c-x-i)/2 for
  1-cx-i and 0 for x-igt1-c
• If we plot the best-reply function for both firms
  in a diagram, we get the following picture

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Example Quantity Competition
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Example Quantity Competition

• The best-reply correspondences are down-ward
  sloping since each firm wants to produce less if
  the competitor is producing more quantities are
  strategic substitutes
• In order to find the NE we solve
  xAlpha(1-xBeta-c)/2 xBeta (1-xAlpha-c)/2
• This gives the NE xAlpha xBeta(1-c)/3

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Example Quantity Competition

• This game is symmetric since uAlpha(xAlpha,xBeta)
  uBeta(xBeta,xAlpha), implying identical
  best-reply correspondences for both firms
• Moreover, from the first-order condition we see
  that both firms must produce equal amounts in a
  NE since xix-i1-(x-ixi)-c
• We can use this fact to solve for an NE in a
  convenient way xi1-2xi-c gives xi(1-c)/3,
  implying a price of p(x)(12c)/3
• Compare this to the monopoly price of p(1c)/2
  and the competitive price of pc

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Example Quantity Competition

• Using this technique we can easily solve for the
  NE in a market with n identical firms of the same
  type
• Denoting total quantity by x, we can write the
  first-order condition for firm is profit
  maximization xi1-x-c
• Since all firms will produce equal amounts in a
  NE this means xi1-nxi-c or xi(1-c)/(n1) for
  all i1,,n
• This gives a price of p(x)(1nc)/(n1), that
  converges to c as n goes to infinity

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Conclusion

• Game theory is a tool for modeling strategic
  interaction
• A strategic-form game consists of the players,
  the players strategies, and their payoff
  functions
• Ways of predicting play
• Iterated elimination of dominated strategies
• Nash equilibrium, strict Nash equilibrium
• Nash equilibrium in mixed strategies
• The Nash equilibria can be calculated using the
  best-reply correspondences