By: Ahmet Kerem Kaymakioglu Sezer Yksek Can Tolga rekioglu

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ByAhmet Kerem KaymakçiogluSezer YüksekCan
Tolga Çörekçioglu
EVOLUTIONARY GAME THEORY
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ADVANTAGES OF EVOLUTIONARY GAME THEORY

• It specifies the stable outcome
• It provides with an explicit process by which the
  stable state is reached
• The evolution of a rationality must be reached
• The evolutionary process considers the survival
  of sub-optimal strategies in equilibrium
• It may prove useful in eliminating the multiple
  equilibria problem

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EVOLUTIONARY STABLE STRATEGIES

• ESS is an incumbent strategy that establishes a
  steady state in a population
• It is immune to invasion by some other mutant
  strategy that occur at a sufficiently smaller
  rate
• The selection process will eliminate the mutant
  by favoring the incumbent

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SYMMETRIC GAMES

• For a two player symmetric game the set of
  the pure strategies is S Si,.........,Sn
• Let a be the mixed strategy ( pi,........,pn
  )
• ? (a, a ) is the expected pay-off to the mixed
  strategy a when faced with another mixed strategy
• a.

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ASSUMPTIONS

• Players from the population are randomly matched
  to play the game.
• There is a small number of players with mutant
  genes that are inclined to play a and the rest
  are inclined to play a.

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Then,

• 1- a is an ESS if
• ? (a , ?a(1- ?) a ) gt ? (a, ?a(1- ?) a )
  
• where ? is the fraction of mutants in the
  population. As we see above the payoff to the
  incumbent strategy is greater than the payoff to
  the mutant strategy when faced with population
  mixture.

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• 2- a is an ESS if
• i) ? (a , a) gt ? (a , a) NE condition
• ii) ? (a , a) ? (a , a) then ? (a , a)
  gt ? (a , a)
• Stability condition
• All ESS are NE, but not all NE are ESS.

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For an application of ESS to a market game
consider the following example.

• The definition of the game is as follows
• 1- Two firms are randomly selected from a
  population of firms to compete in a market.
• I firm1 , firm2.
• 2- Each player has two available strategies as
  accommodate or fight the opponent. Then
• S accommodate , fight.

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• 3- The entries in the matrix show the payoffs to
  both players.

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SOLUTION

• Suppose that f is smaller than d and a is
  smaller than w.
• Then, the pure strategy nash equilibria are
  fight,accommodate and accommodate,fight.
• Now, we need to find the ESS.

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SOLUTION

• A population consisting of accommodating firms is
  vulnerable against fighters. The expected pay-off
  for an incumbent firm is a, since it will face
  its own type. The rare fighter, on the other hand
  earns a higher pay-off, w.

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SOLUTION

• Since the intruder strategy has a greater
  pay-off, the firms playing the strategy fight
  increase over time. This is due to some close
  observer incumbent firms imitating the more
  successful strategy and switching from
  accommodate to fight. Hence, the
  accommodating population is invaded by the
  fighters.

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SOLUTION

• There is also a mixed strategy equilibrium a,
  a, that assigns following probability to pure
  strategy fight
• P (a-w)/(f-da-w)
• Then, each firm assigning P to fight, and
  (1-P) to accommodate is the unique mixed
  strategy equilibrium.

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SOLUTION

• The symmetric mixed strategy profile (a ,
  a) P,1-P)(P,1-P) constitutes an ESS if
  the following condition is satisfied for all a is
  not equal to a
• ?(a, a)gt ?(a, a).
• Where ?(a, a) represents the pay-off
  function to the mixed strategy a when faced with
  the mixed strategy a.

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SOLUTION

• Likewise, ?(a,a)is the pay-off to mixed strategy
• a when faced by itself. Hence, the condition
  implies
• that a must be a better response to a, then a
  to
• itself. Then, ?(a,a) - ?(a,a) gt 0.
• Hence, checking the above condition yields
• P. P.f (1-P).w (1-P).P.d (1-P).a
  
• PP.f (1-P).w (1-P). P.d (1-P).a

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SOLUTION

• ?(P-P).P.f (1-P).w (1-P - 1 P).P.d
  (1-P).a
• ?(P-P).P.f (1-P).w (-PP).P.d
  (1-P).a
• ?(P-P).P.f w- P.w - P.d a P.a
• ?(P-P).P(f w d a) w a.

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SOLUTION

• Given P (a-w)/(f-da-w), we have
  (w-a)-P.(f-da-w) and substituting it in
• (P-P)².-(f-w-da)
• ?(P-P)².-fwd-a) gt 0, since altw and fltd.
• Therefore, the mixed strategy a is an
  ESS.Hence, the established strategy a will
  resist intrusions by any other entrant strategy.

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INTERPRETATION

• A population with a proportion of P fighting and
  (1-P) accommodating firms is immune to
  intruders.
• Note that this particular game assumes identical
  firms with symmetric action sets and does not
  allow for strategic interaction.

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ASYMMETRIC GAMES

• In most situations economic games are not
  symmetric.
• Asymmetries can arise due to different action
  sets available to each player, because
• -They come from different populations
• or
• -They are of different types in the same
  populations.

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ASYMMETRIC GAMES

• Consider the finite two player game GI,A, ?
  and its mixed strategy extension (I,T,u) for a
  formal definition.
• I, is the set of players
• A, is the set of pure strategies
• , is the set of mixed strategies
• u, is the set of pay-offs assigned to mixwd
  strategies

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ASYMMETRIC GAMES

• Pairs of players are selected randomly from a
  large population to play the game, and also being
  selected the players can observe the type they
  are supposed to play. This makes it possible for
  the players to condition their strategies on the
  type they observe. Note that the probability of
  individuals to be selected to play any of the two
  types is identical for all individuals.

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ASYMMETRIC GAMES

• Then the behavioural strategy is a pair of mixed
  strategies, a (a1,a2) ? T .
• a1, is used in row players position and
• a2 is used in column players position.
• The pay-off matrices for row and column players
  are M and N, respectively.

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ASYMMETRIC GAMES

• The expected pay-off to behaviour strategy a ? T
  when it is played against the strategy ß
  (ß1,ß2) ? T is given by
• u(a,ß) ½ u1(a1,ß2) u2(ß1,a2)
• ½a1M ß2 ß1N a2.
• Hence, a is called an ESS if it is a best reply
  to itself.

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ASYMMETRIC GAMES

• In other words
• u(a,a) gt u(ß,a), for ß .
• If the first condition is satisfied with
  equality, then it must be the case that
• u(a,ß) gt u(ß,ß), for all ß .
• ? If a behaviour strategy of the generated
  extensive form game is ESS, then it is also a
  strict nash equilibrium of the underlying game G.
  

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ASYMMETRIC GAMES

• Then
• ½ u1(a1,a2) u2(a1,a2) gt
  ½u1(ß1,a2) u2(a1,ß2).
• And, by including the pay-off matrices and
  multiplying each side by two, we have
• a1M a2 a1N a2 gt ß1M a2 a1N ß2.
• This allows us to apply the ESS conditions to
  asymmetric games.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• The definition of the game is as follows
• 1- Consider two firms competing in this two
  period entry deterrence game. Hence the set of
  players is I Firm1 , Firm2 . Firm1 is the
  incumbent firm and Firm2 is the potential
  entrant. Firm2 is the first mover and the Firm1
  responds in the second period.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• 2- In the first period, the action set of the
  potential entrant is A2 enter , stay out . The
  incumbents action set is A1 fight ,
  accommodate .
• 3- The entries in the matrix show the payoffs to
  the row player and to the column player
  respectively.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE
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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• In this game, there is a unique subgame
  perfect equilibrium, where the entrant enters and
  the incumbent accommodates. We can view this
  equilibrium as a degenerate mixed strategy Nash
  equilibrium, where the entrant enters with
  probability one and the incumbent accommodates
  with probability one.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• For simplicity lets use the matrix form
  representation and denote the payoff matrices of
  the incumbent and the entrant with I and E,
  respectively. Then we have,
• I 0 4 E 0 1
• 2 4 2 1

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• For convenience we can transform the matrices
  by subtracting 4 from the second column of I, and
  dividing all entries of I by 2. We get
• I 0 0
• 1 0
• Then subtracting one from the second row of E
  we get
• E 0 1
• 1 0
  

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• Consider the behavior strategy
• a (a1,a2) ? T, where mixed strategies a1 and
  a2 is to be used when in the position of the
  incumbent and the entrant respectively.
• Suppose that a1(a2) assigns a probability of
  P1(P2) to the entrant playing the pure strategy
  Enter(Fight).

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• As a result, a1(a2) assigns a probability of
  1-P1(1-P2) to the entrant playing the pure
  strategy stay out.
• Consider an alternative behavioral strategy
  ß(ß1,ß2) ? T. Suppose that ß1(ß2) assigns a
  probability of Q1(Q2) to the entrant playing the
  pure strategy enter(fight). As a result, ß1(ß2)
  assigns a probability 1- Q1(1-Q2) to the entrant
  playing the pure strategy stay out.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• Then for the behavioral strategy aa1(a2) ? T
  to be ESS it must satisfy the following condition
  for all ß(ß1,ß2) ? T, ß is not equal to a. So,
• a1 I a2 a1 E a2 gt ß1 I ß2 ß1 E
  ß2
• ?(1-P1)P2 0 (1-P1)P2 (1-P2)P1gt (1-Q1)P2
  0 (1-P1)Q2 (1-Q2)P1
• ? P2(1- 2P1)-P1 P2 gt Q2 (1- 2P1)-Q1 P2.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• Let us consider the candidate strategies for
  ESS, starting with the mixed strategy set P10,
  P2gt½. Such mixed strategies can not be ESS in
  an asymmetric game. When P2 component of this
  strategy mutates to any value above ½, it implies
  that the entrant mixed strategy will constitute
  an alternative best reply to the mixed strategy
  in concern. There are several best replies, since
  the mixed strategy involves a continuum which
  violates the strict Nash equilibrium.

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ASSYMMETRIC ENTRY DETERRENCE EXAMPLE

• Hence, our only candidate is the unique subgame
  perfect equilibrium enter, accommodate of the
  base game, where the entrant always enters and
  the incumbent always accommodates. This pure
  strategy equilibrium corresponds to the
  degenerate mixed strategy (1,0),(0,1), which
  satisfies the inequality condition.

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REPLICATOR DYNAMICS

• Lets suppose that pairs of players are
  repeatedly selected from a large but finite
  population to play a symmetric normal form game.
  Every member of the population is programmed to
  play a pure strategy ai ?
• Ai ( i1,2,....,k).At any point t in time let
  Ni (t) be the number of players in the population
  that play the pure strategy ai and N(t) be the
  total number of players in the population.

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REPLICATOR DYNAMICS

• Then we can define the population share of
• the ai players as pi(t) Ni(t) / N(t) .
  Therefore, p(t) is similar to a mixed strategy in
  the mixed strategy simplex ? .
• Then the expected payoff to any pure strategy ai
  may be denoted by ?(ai,p). Here we see that the
  average payoff in the population becomes the
  payoff to the mixed strategy p when it is matched
  by itself.

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REPLICATOR DYNAMICS

• The equation below gives this average payoff in
  the population
• k
• ?(p,p)? pi ?(ai , p)
• i1

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REPLICATOR DYNAMICS

• Assuming continuous time for convenience, the
  replicator dynamics first introduced by Taylor
  and Jonker is as in
• ?pi / ?p?(ai , p)- ?(p , p) pi .
• Under linear payoffs we can rewrite this equation
  as
• ?pi / ?t?(ai-p , p) pi .

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REPLICATOR DYNAMICS

• Hence, the population shares of the strategies
  that earn higher than the population average
  payoff increases. Whereas, the population share
  of the strategies faring below the average
  declines. Therefore, the evolutionary dynamic is
  captured by this equation.

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REPLICATOR DYNAMICS

• The biological interpretation of the dynamics and
  its use in biological contexts is straightforward
  where fitness is measured as the number of
  offsprings. The economic justification, however,
  is more challenging. In an economic context, we
  can think about players imitating the successful
  strategy that yields a higher payoffs than their
  own strategy.

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REPLICATOR DYNAMICS

• Therefore, the number of players using the less
  profitable strategy declines and the number of
  players using strategies that earn above average
  increases over time. It may be assumed that the
  players are myopic, which means they do not take
  into account the future developments in the
  population. Furthermore we can assume that
  players do not change their strategies frequently.