Fictitious Play The Theory of Learning in Games D. Fudenberg and D. Levine

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Fictitious PlayThe Theory of Learning in
GamesD. Fudenberg and D. Levine

• Speaker Tzur Sayag
• 03/06/2003

2
Do you believe that PM Sharon is serious about
the peace process?

• A voter has to decide if he should support PM
  Sharon
• Belief Sharon will never evacuate settlements
• Action Vote against the new economics
  revolution.
• May 24 Sharon announces occupation is no-good
• Belief Sharon will probably never evacuate
  settlements
• Action Vote against the new economics revolution
• Jun 5 Sharon meets Abu-Mazen and declares
  support for a Palestinian state.
• Belief Seems like Sharon might evacuate the
  settlements after all
• Action Vote for the new economics revolution.

3
Roadmap

• Introduction to the common models of learning in
  games
• Cournot adjustment
• Fictitious play and Nash equilibriums
• Motivation
• Definitions
• Results
• Generalizations of fictitious play if we have
  time

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Notations
P1 gets a1 and p2 gets b1 if they play
Action1,Action1 respectively
Player 2
Action1 Action2
Action1 (a1,b1) (a2,b2)
Action2 (a3,b3) (a4,b4)
Player 1
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Learning in Games - 1

• Repeated games same or related
• fixed-player model
• Teach the opponent to play a best response to a
  particular action by repeating it over and over.

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Being Sophisticated Example

• D is dominant for Bob.
• If Alice learns Bob only plays D, game converges
  to ltD,Lgt
• Bobs payoff for ltD,Lgt is 2.
• If Bob is patient, he can play U always and just
  wait for a while
• If Bob always plays U,
• Alice who thought Bobs gonna play D should shift
  its play from L to R (since R was only good when
  Bob actually played D)
• So Bob plays constant U which leads Alice to play
  constant R with payoff 2 gt 1.
• in this case Bob gets 3 which is better.
• Bingo!

Alice
L R
U 1,0 3,2
D 2,1 4,0
Bob
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Being Sophisticated Abstracting

• Most learning theory rely on models in which the
  incentive is small to alter the future play of
  the opponent.
• Locked in for 2 periods
• Large anonymous population
• Embed a two player game by pairing players
  randomly from a large population.

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Models of Embedding

• Single-pair model
• random single pair, actions revealed to everyone
• Aggregate static model
• all players randomly matched, aggregates outcomes
  revealed to everyone
• Random-matching model
• all players randomly matched, each player sees
  his game outcome only

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Three common models of Learning

• Fictitious play
• Players observe only their own matches and play a
  best response to the frequencies.
• Partial best-response
• A fixed portion switches each period from its
  current action to a BR to the aggregate stats
  from the previous period.
• Replicator Dynamics
• The share of the population using each strategy
  grows proportionally to that strategys current
  payoff.

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Cournot Adjustment a flavor of analysis

• Two firms 1 and 2.
• Strategy choose a quantity si?0,8)
• Strategy profile is ltsi, s-igt?S
• Utility for i is ui(ltsi, s-igt)
• Assume ui(lt., s-igt) is strictly convex
• BR(s-i) argmax ( ui(ltx, s-igt) ) x?S

BR is unique since u is concave so the relevant
u is positive, this means that u is a monotone
increasing function which means it has at most
one zero which means, yes, you guessed it right U
only has one extreme point and the max is
therefore unique. u cant be fixed since it is
STRICTLY concave by assumption
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Cournot Adjustment Model

• time periods t 0,1,2,, discrete
• State profile ?0 ?S
• in each period the player chooses a pure strategy
  that is BR to the previous period
• Formally i chooses stBR(s-it-1)

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Cournot DynamicsReaction Curve
BR1 For every ?2 the line states the BR of player
1 against it. The value for player 1 is the
height at point ?2
?t (?t1 , ?t2)
?2
Can you convince yourself this point is a Nash?
?t1
?t2
BR2
?1
?t1
New BR if 2 plays ?t2
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Cournot Dynamics

• A movement between profiles such that
• ?t1 f(?t) , fi(?t) BRi(?t-i)
• A steady state is ?s s.t. ?s f(?s)
• Once ?t ?s the system remains there
• Claim (simple) ?s is a NASH
• Proof by definition for every player
  ?sBRi(?-i), so players dont want to move.
• SO EVERY STEADY STATE IS A NASH EQUILIBRIUM

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Cournot Dynamics oblivions to linear
transformation

• Proposition 1.1 Suppose ui(s)aui(s) vi(s-i)
  for all players I, Then u and u are
  best-response equivalent
• Proof
• vi(s-i) is dependent on the opponents play so it
  does not change the magnitude order (seder)
  of my actions
• Multiplying all payoffs by the same constant a
  has no effect on the order
• So, a transformation that leaves preferences, and
  consequently best responses, will give rise to
  the same dynamic learning process.

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Cournot Dynamics and Zero sum Games

• Recall payoffs in ZSG add to zero.
• Proposition 1.2 every 2 x 2 game for which the
  best response correspondences have a unique
  intersection that lies in the interior of the
  strategy space is best-response equivalent to a
  zero-sum game.
• Proof given G, a 2x2 game, with unique
  intersection,
• w.l.o.g. assume 1) A is BR for player 1 against
  A 2) B is BR for player 2 against A
• If A was also a BR for player 2 then ltA,Agt is a
  BR correspondence at a pure profile which
  contradicts our assumption.

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Cournot Dynamics and Zero Sum Games 2

• Proof outline Given G, the 2x2 game with unique
  intersection, we build a zero sum game that has
  the same Best Responses. Observe the following
  zsg.
• If alt1 then BR1(A)A since u1(A,A) 1 but
  u1(A,B) is only a
• If alt1 then BR2(A)B since u2(A,B) 0 but
  u1(A,A) is only 0
• Denote si player is probability to play A
• Claim 1 player 1 is indifferent between A and B
  if, s2 a s2 b (1- s2)
• Claim 2 player 2 is indifferent between A and B
  if, s1 a (1-s1) b (1- s1)

A B
A (1,-1) (0,0)
B (a,-a) (b,-b)
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Proof ofplayer 1 is indifferent between A and B
if, s2 a s2 b (1- s2)

• Assume s2 a s2 b (1- s2) () (s2 is the
  prob. 2
• (1) If player 1 plays A he (1) gets plays A)
• u1(A,?) s2 (u1(A,A)) (1 - s2)
  (u1(A,B))? s2 (1) (1- s2) (0) s2 (by
  the game table)
• (2) If player 1 plays B he gets
• u1(A,?) s2 u1(B,A) (1 - s2)
  u1(B,B)? s2 (a) (1- s2) b (by the game
  table)
• So if (1) (2) he does not care which to choose,
  (1) (2)? s2 s2 (a) (1- s2) b as
  required.
• Proof of claim 2 regarding 2s indifference
  follows the same path.

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Proof contBuilding the ZSG Game
Mental note si Prplayer i playing A

• 1 is indifferent between A and B if s2 a s2
  b (1- s2)
• 2 is indifferent between A and B if s1 a
  (1-s1) b (1- s1)
• Fixing an intersection point s1, s2 We can solve
  for the unknown payoffs a,b a (s2 s1) / (1
  s2 s1) Notice that (s2 s1) lt 1 (si gt 0
  otherwise i never plays A)
• (s2 s1) lt 1 implies alt1 (since (1 s2 s1)gt1)
  Q.E.D.We already showed that when alt 1 it means
  that we get the same best responses we had in the
  original game G A for player 1 against A, B
  for player 2 against A
• To sum up it should have been obvious that (s1,
  s2 ) is a Nash, the point was to find a 2x2 ZSG
  which has the same best responses as the original
  game

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Strategic-Form Games

• Finite actions
• One shot simultaneous-move games
• Players, strategy space, payoff functions is
  the strategic form of a game

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Nash and Correlated Nash

• A game can have several NashsltA,Agt,
  ltB,Bgt,lt(1/2,1/2), (1/2,1/2)gtbut the payoffs may
  be different.ltA,Agt gets 2 for eachlt(1/2,1/2),
  (1/2,1/2)gt gets 1 for each.
• Lets question the robustness of the mixed
  strategy Nash point.
• Intuitively, at the mixed, players are
  indifferent (in real life) play A,B whateverso
  one may believe that the other one plays A with
  slightly more probability. He then wants to
  switch to pure A so the robustness of Nash seems
  questionable..

A B
A (2,2) (0,0)
B (0,0) (2,2)
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Nash and Correlated Nash

• A Nash is strict if for each player i, si is the
  unique best response to s-i
• Only pure strategies can be strict since if a
  mixed is BR than so is every pure strategy in the
  mixed strategys support otherwise there is no
  point of including it.
• Recall Support for a mixed strategy are the pure
  strategies that participate with positive
  probability.

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Some Questions in Theory of Games

• When and why should we expect play to correspond
  to a Nash equilibrium
• If there are several Nash equilibria, when one
  should we expect to occur?
• In the previous example, in the absence of
  coordination, we are faced with the possibility
  that player 1 expects NE1ltA,Agt so he plays A,
  the opponent might expect NE2ltB,Bgt and he plays
  B, with the results of the non-equilibrium
  outcome profile ltA,Bgt

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The Idea of Learning based explanation of
equilibrium

• Intuitively, the history of observations can
  provide a way for the players to coordinate their
  expectations on one of the two pure-strategy
  equilibrium.
• Typically, Learning models predict that this
  coordination will eventually occur, with the
  determination of which of the two eq. arise left
  to initial conditions or to random chance.

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The Idea of Learning based explanation of
equilibrium

• For the history to serve this coordination role,
  the sequence of actions played must eventually
  become constant or at least readily predictable
  by the players, of course, there is no
  presumption that this is always the case.
• Perhaps, rather than going to a Nash, players
  wander around the space aimlessly, or perhaps
  play lies in some set of alternatives larger than
  the set of Nashs?

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The Idea of Learning based explanation of
equilibrium

• For the simple coordination game (symmetric
  lt2,2gt, lt0,0gt) there is no reason to think that
  any learning process will prefer one Nash over
  the other.
• What if we alter it such that there is a better
  Nash. Will the players learn to play the ltA,Agt
  Nash?

Altered
(2,2) (-a,0)
(0,-a) (1,1)
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Correlated Nash (Aumann 74)

• Suppose the players have access to randomized
  devices that are privately viewed.
• If a player chooses a strategy according to his
  own randomized device, the result is a
  probability distribution over strategy profiles,
  denoted µ??(S).
• Unlike a profile of mixed strategies which is by
  definition uncorrelated, such a distribution may
  be correlated.

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Correlated Nash Jordans matching pennies

• 3 players.
• Each chooses H or T
• Payoffs are 1 or -1 only
• 1 wins if he matches 2
• 2 wins if he matches 3
• 3 wins by not matching 1
• This game has a unique NE, each play (1/2,1/2)
• HoweverIt has many correlated NE.

Player 3 plays H
H T
H 1 ,1,-1 -1,-1,-1
T -1, 1, 1 1 ,-1, 1
Player 3 plays T
H T
H 1 , 1, -1 -1,-1,-1
T -1, 1, 1 1, -1, 1
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Correlated Nash Jordans matching pennies
Player 3 plays H

• C-NE unified distribution over these 6
  profiles(H,H,H) (H,H,T) (H,T,T) (T,T,T)
  (T,T,H) (T,H,H)
• Each player has 50 to play H.
• No weight is placed on (H,T,H), so the play of
  the players is not independent (it is correlated)
• For Player 1 When he plays H he faces 1/3 chance
  each of his opponents play (H,H), (H,T),(T,T).
  Since his goal is to match 2, he wins 2/3 of the
  times by playing H and only one third if he plays
  Y. similarly if he plays T his opponents might
  only play (T,T), (T,H), (H,H). Now tails win 2/3
  of the times as against heads which wins only 1/3
  of the time. So he is evened. He is at a Nash.

H T
H 1 ,1,-1 -1,-1,-1
T -1, 1, 1 1 ,-1, 1
Player 3 plays T
H T
H 1 , 1, -1 -1,-1,-1
T -1, 1, 1 1, -1, 1
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Why is Correlated Nash of Significance?

• Hint Cycles create correlation between profile
  strategies.
• Informally a cycle is a finite sequence of
  profiles of length k such that s0sk.
• Cournot play can exhibit cycles example
  follows.
• So cycles gt correlation gt correlated Nash

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Cournot Cycle - matching pennies.

• 3 player (head, tail)
• 1 wants to match 2.
• 2 wants to match 3.
• 3 wants to un-match 1.
• Cournot means each player assumes his
  opponents play the same as in their last step

Current Profile(s1,s2,s3) Remarks
H,H,H P3 H-gtT (switch) P2 H-gtH (stay) P1H-gtH (stay)
H,H,T P2 H-gtT
H,T,T P1H-gtT
T,T,T P3 T-gtH
T,T,H P2 T-gtH
T,H,H P1T-gtH
H,H,H P3 H-gtT
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Roadmap

• Introduction to the common models of learning
• Cournot adjustment
• Fictitious play and Nash equilibriums
• Motivation
• Definitions
• Results
• Generalizations of fictitious play

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Fictitious play - Introduction

• Motivation
• Repeated game, stationary assumption.
• Each player forms a belief of his opponents
  strategy by looking at what happened
• Player plays Best Response according to his/her
  belief

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Two-Player Fictitious Play - notations

• S1 and S2 are finite actions spaces for players
  one and two respectively.
• S1 ,?,?
• S2 ?,?,?
• u1, u2 player payoff functions
• u1(, ?)15
• for mixed strategy we take
• u1(lt½,½gt,lt¼, ¾gt ) u1(
• u1(,lt¼, ¾gt )¼ u1(, ?) ¾ u1(, ?)
• Player is pi, opponent is p-i i1,2

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Two-Player Fictitious Play

• Notion of belief
• A prediction of the opponent action distribution
  the degree to which 1 believes 2 will play ?
• Assume players choose their actions for each
  period to maximize their expected payoff, with
  respect to their belief for the current period.

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Two-Player Fictitious Play Forming Beliefs

• Player i starts with a weight function K0i
• K0i S-i ? ?
• For example
• K0i?,,? ? ?
• K0i()4
• As the game is iteratively repeated K is updated

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Two-Player Fictitious Play Belief update

• If some action say was played (by the
  opponent!) the last time, we add 1 to its count,
  generally
• Kt (s-i) 1 if s-it-1 s-i
  0 otherwise
• Thats a complicated way of saying that K(s)
  simply counts the number of times the opponent
  played s.

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Two-Player Fictitious Play Using frequencies to
form beliefs

• Given K the frequency vector,
• Each player forms a probability vector ? over his
  opponents actions
• His belief can be said to be that the
• Pri plays Kt() / steps
• Simple normalization

Reads the belief player i holds at time t
regarding the probability of his opponent to
plays s-I in time t
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Two-Player Fictitious Play Using frequencies 2
My belief is that my opponent plays ? with
probability ½, ,? with prob ¼ and ¼ ?, looking at
my payoff table, by playing I can max the
utility

• We now have a belief of how the opponent plays.
• A FP is any rule ?it which assigns a Best
  Response action to the belief ?it
• Example
• ?1(lt½,¼,¼gt) (extend naturally to mixed)
• This implies that u1(, lt½,¼,¼gt) is better
  for player 1 than any other action against
  lt½,¼,¼gt

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Two-Player Fictitious Play remarks

• Many BR are possible for a given belief set
• An example of such rules ? may be
• Always prefer pure action over mixed action
• Pick the best response for which your action
  index is least, (thats the limit of my
  creativity)
• (both of course must still be best responses)

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Two-Player Fictitious Play Interpretation (page
31-32)

• Bayesian inference
• Player i believes opponents play corresponds to
  a sequence of i.i.d. multinomial random variable
  with a fixed but unknown distribution.
• Player is prior over that unknown distribution
  takes the form of a Dirichlet distribution.
• is prior and posterior belief corresponds to a
  distribution over the set ?(S-i) of probability
  distributions over S-i
• The distribution over oppnents strategies ?I t
  is the induced marginal distribution over pure
  strategies.
• If beliefs over ?(S-i) are denoted µi, then we
  have

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Two-Player Fictitious Play Interpretation

• Denote the marginal empirical distribution as
• The assessment ? is not the same as d because of
  the influence of is prior belief
• This has the form of a fictitious sample
  observed before the game started.
• As observations are incorporated into ?, it will
  converge to d (the empirical distribution)

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Two-Player Fictitious Play Interpretation

• Notes
• As long as the initial weights are positive it
  will stay positive
• The belief reflects the conviction that the
  opponent strategy is constant and unknown.
• It may be wrong If the process cycles.
• Any finite sequence of what looks like a cycle is
  actually consistent with this assumption that the
  world is constant and those observations are a
  fluke
• If cycles persist, we might expect i to notice it
  but in any case, his beliefs will not be
  falsified in the first few periods as they did in
  the Cournot process.

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Asymptotic Behavior does play converges

• Sufficient conditions
• Proposition 2.1
• (1) if s is a strict Nash and is s is played at
  time t in the process of FP then s is played at
  all subsequent dates
• (2) any pure-strategy steady state of FP must be
  a Nash

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If s a strict Nash and played at time t s is
played at all subsequent dates

• Proof
• Suppose ?it (players beliefs) are such that the
  actions are strict Nash s.
• believe me that When profile s is played at
  time t, each players belief at t1 are a convex
  combination of ?it and a mass point on s-i ?it1
  (1-at) ?it atd(s-i)
• we get

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If s a strict Nash and played at time t s is
played at all subsequent dates

• We want to show that this payoff is still better
  than any other payoff involving ?it1
• Now si was a strict BR for ?it
• Should be obvious for the first term (by
  assumption that it is strict BR for ?it).
• for the second term, note that for the point mass
  it is obvious that si is better because it
  implies that the profile lt si , s-i gt is a Nash
  which was our assumption

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So, what is a point mass on s-i and why is ?it1
a convex combination of it and of ?it ?

• I need to show you that ?it1 (1-at) ?it
  atd(s)
• For clarity, lets say t10, there are 2 players,
• Lets say s( ½, ½ ), ( ¼ , ¾ )
• S-iS2C,D and look at ?1101(C)
• recall ?110 (C)K10 (C) /10 (ignore prior it
  matters not)
• Suppose that at time 10 player 2 actually played
  C (he played a mixed which is interpreted that ¼
  of the times he would play C..)
• ?111 (C) K10 (C) 1 / 11

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2nd part Any pure-strategy steady state of FP
must be a Nash

• A steady state is a strategy profile that is
  played in every step after perhaps a finite time
  T.
• Ideas?
• If play remains at a pure-strategy profile then
  eventually the assessments will become
  concentrated at this profile.
• If it was not a Nash for one of the players, him
  playing what he played would not be a BR, this is
  a contradiction to how FP works,
• Since all players always play BR according to
  their belief.
• Food for thought Why does it not work for
  mixed-strategy profile?

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To Conclude this

• we wanted to show that if s is a strict Nash and
  is s is played at time t in the process of FP
  then s is played at all subsequent dates
• We showed it by looking at what happens to
  players belief and prove that the actions at
  given the new belief are still strict BR.
• This means the system is at a steady state.
• We also showed that if it is a pure-strategy
  steady state it is a Nash.

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No Pure Nash gt FP cant converge to a pure
profile

• Matching pennies
• For example
• At time3 player I believes that II prefers
  Tails, so he plays Tails to match
• But II plays Heads so I adds one to Heads
• Now HeadsgtTails and I convinced himself II will
  play Heads so he switches to H
• The game cycles and never converges to the Nash
  profile.

H T
H 1,-1 -1,1
T -1,1 1,-1
T Profile I II
1 Initial (1.5,2) (2,1.5)
2 (T,T) (1.5,3) (2,2.5)
3 (T,H) (2.5,3) (2,3.5)
4 (T,H) (3.5,3) (3,3.5)
5 (H,H) (4.5,3) (4,3.5)
50
No Pure Nash gt FP cant converge to a pure
profile

• If the game did converge it would be in a
  steady state that is pure and not a Nash (since
  matching pennies has no pure Nash) but we showed
  that any pure-steady state must be a Nash.
• Its ok then that the game does not converge.
• Interestingly, the empirical distributions over
  player is strategies are converging to ( ½ , ½ )
  their product ( ½ , ½ ), ( ½ , ½ ) is a Nash.

51
Asymptotic Behavior

• Proposition 2.2
• If the empirical d over each players choices
  converges, the strategy profile corresponding to
  the product of these distributions is a Nash
• Proof
• intuitively, if the empirical does converge, then
  the belief converges to the same thing, hence, if
  it was not a Nash players would move from there.
• Generally, for this it is enough that the beliefs
  are asymptotically empirical, need not be FP

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Asymptotic Behavior

• More results (proof omitted)
• The empirical converges if
• (1) generic payoff and 2x2 game
• (2) zero sum
• (3) solvable by iterated strict dominance
• The empirical distribution however need not
  converge! 2 examples.

53
Example Shapley (1964)
L M R
T (0,0) (1,0) (0,1)
M (0,1) (0,0) (1,0)
D (1,0) (0,1) (0,0)
Nash is at ( 1/3, 1/3,1/3) for both. if initial
weights lead to ltT,Mgt we cycle. Diagonals are
never played. (ltT,Lgt, ltD,Mgt,ltT,Mgt the number of
consecutive periods each profile is played
increases sufficiently fast so the empirical
distributions never converges.
54
Example due to Jordan

• Increase to (1/2,1/2) the diagonal.
• We get Even-Nyar Umisprayim a zero sum game
  with the same Nash
• Here, since the empirical do converge, there are
  still cycles, but the rate of repeated profiles
  within a sequence grows slowly.
• Note that this does not conflict with our
  previous statements, namely
• Heres a zsg whos empirical converge but its
  not steady (cycles)
• Had the empirical converged to a pure-strategy
  Nash the fact that it cycled would have been a
  conflict to ltLINKgt

55
Payoffs in Fictitious Play

• The question we deal with here is if FP learns
  the distributions then it should, asymptotically
  yield the same utility that would be achieved
  when the frequency distribution is known in
  advance.
• Here we will suppose more than 2 players, their
  assessment track the joint distribution of
  opponent strategies.

56
Payoffs Fictitious Play - Notations

• Empirical joint distribution
• Best payoff against the empirical
• Time avg of realized payoffs
• Definition
• Fictitious play is e-consistent along a history
  if there is T such that for any T

57
Payoffs Fictitious Play - Notations

• Empirical joint distribution
• Best payoff against the empirical
• Time avg of realized payoffs
• Definition
• Fictitious play is e-consistent along a history
  if there is T such that for any T

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Consistency

• We dont look at how good the player does
  globally but how good he does with comparison to
  his expectations which are built upon his
  beliefs.
• If the FP is not consistent it would be much less
  interesting model, it would be as if someone
  simply plays a different game. So, we want
  consistency.

59
Consistency

• If A game is consistent it can be useful, if
  after some period of time, a player sees that his
  expectations are not fulfilled (by comparing the
  expected payoff with the actual payoff), he can
  deduce that something is wrong in his model of
  the world.

60
Consistency main result

• Relate player frequency of switching strategies
  of a player to the consistency of his play.
• We define the frequency of switches ?it to be the
  fraction of periods for which

61
Fictitious Play and BR dynamics

• Definition
• Fictitious play exhibits infrequent switches
  along a history if for every e gt 0 there is a T
  and for any t ? T, ?it ? e for all i.
• Proposition (Fudenberg and Levine 94)
• If FP exhibits infrequent switches along a
  history, then it is e-consistent along that
  history for every e gt 0

62
infrequent switches ? e-consistency

• Intuition
• Once prior looses its influence, at each date
  player i plays BR to the empirical (observations)
  through date t-1.
• On the other hand, if i is not doing on average
  as well as best response to the empirical, there
  must be a nonmalleable fraction of dates t for
  which i is not playing BR, but in this case,
  player i must switch in date t1. conversely,
  infrequent switches imply that most of the time,
  is date-t action is a best response to the
  empirical distribution at the end of date t.

63
Proof - notations

• k length of initial history (kept as prior
  belief)
• initial belief
• best response strategy of i to
• player i expected date-t payoff is

64
Proof - summary

• We showed that if there are not many switches
  along the players history, his play is
  consistent.
• It means that the actual payoffs do not flounder
  below the expectation!
• Again, one can use this to see if something is
  wrong with his model of the env.

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Roadmap

• Crash introduction to the common models of
  learning
• Cournot adjustment
• Fictitious play and Nash equilibriums
• Motivation
• Definitions
• Results
• Generalizations of fictitious play

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Generalizations of Fictitious Play

• Mainly generalize the update rule for the belief
• Example exponential weighting
• As long as beliefs are asymptotically empirical
  what we showed for FP holds.

67
Summary

• Dynamics of Games and a flavor of analysis
• Although different from Nash analysis, Nash is
  still an important point, we showed that it is
  still a point FP can converge to
• FP is a consistent play
• Have a nice summer vacation thanx for listening.