Game Theory Folk Theorems

1
Game Theory Folk Theorems

• Univ. Prof.dr. M.C.W. Janssen
• University of Vienna
• Winter semester 2008-9
• Week 48 (November 24-6)

2
Notation in repeated games

• Define history of play as follows.
• Let a0 (a01 , a02 ,,a0n ) as the action
  profile that is played in stage 0, i.e., the
  actions played by all players
• History at the beginning of period 1, h1 a0
• History at the beginning of stage t1, ht1
  (a0,,at)
• The set Ht is the set of all possible histories
  ht and Ai(ht) is the set of actions that player i
  can choose after history ht and Ai(Ht) is the
  union of this set over all possible histories
• Strategy si of player i is a sequence of mappings
  ski where each ski maps Hk to mixed actions.
• Note that you cannot condition on the random
  events

3
Pay-offs in repeated games

• Overall pay-offs ui stage game pay-offs gi,
  continuation pay-off from period t onwards
• Want to have an expression where one can easily
  compare stage game pay-offs and repeated game
  pay-offs, i.e., normalisation
• Time averaging is sometimes used for the case of
  complete patience

4
Folk Theorem I

• If players are sufficiently patient, then any
  feasible, individually rational pay-offs can be
  enforced by an equilibrium
• Individually rational pay-offs minimax pay-off
  
• vi
• mji is action player j chooses to minimax player
  i
• Feasible pay-offs is the convex hull V of the
  static game pay-offs, i.e., V convex hull v /
  there is an a ? A such that g(a) v
• Both terms need some explanation

5
Minimax pay-offs

• What are the Nash equilibria of this game?
• Pure strategy eq (D,L), (D,R)
• Denote by q the probability player 2 chooses L
• In a mixed strategy eq ?q?, pay-offs also 0 and
  1
• Minimax for player 1
• u(U) -3q1
• u(M) 3q-2
• U(D) 0
• Minimax is 0
• Minimax for player 2 is also 0
• By 1 choosing (½,½,0)
• Thus, minimax poay-offs can be lower than Nash
  eq. pay-offs

6
Feasible pay-offs

• Equilibrium pay-offs are (2,1), (1,2) and (?, ?)
• Convex hull is triangle connecting the three
  points (also e.g. (1½,1½))
• But (1½,1½) cannot be obtained by independent
  mixing, only as correlated eq
• Correlated mixing can happen in repeated setting
  by alternating between playing two equilibria
  (and time averaging pay-offs or d close to 1)

Eq. pay-offs
7
Folk Theorem II

• Prop. For every feasible pay-off vector v with vi
  vi, there exist a d lt 1 such that for all d gt d
  there exist a nash equilibrium of the infinitely
  repeated game with pay-off v.
• Pay-offs in repeated game cannot only be larger,
  but also smaller than static Nash eq pay-offs!!
• Basic idea if players are sufficiently patient,
  then any finite gain in a one period deviation is
  nothing compared to a small, but permanent loss
  in future pay-offs (punishment by minimaxing a
  player)

8
Proof Nash Folk Theorem

• Consider feasible pay-off v and action profile
  g(a)v
• If there is no action profile a that yields v,
  you may choose a sequence of actions such that v
  is (close to) average (discounted) pay-offs (or a
  public randomization)
• Consider strategy start by playing ai play ai
  as long as others do, if one player j deviates
  minimax him forever, i.e., choose mji
• Deviation in period t pay-off yields
• which is smaller than vi if d is larger than di,
  where di solves

9
Is the threat of Minimaxing credible?

• If we restrict analysis to static Nash threats,
  then Friedman shows that only pay-offs larger
  than the static Nash equilibrium pay-offs can be
  supported
• Others show that in games where the minimax
  pay-offs are lower than the static equilibrium
  pay-offs, even worse outcomes can be compatible
  with a SPE of the infinitely repeated game.

10
Basic idea of SPE with minimax pay-offs time
averaging

• After a deviation, play the minimax pay-off for N
  periods, where n is chosen for all players s.t.
• After N periods return back to cooperative mood
• (finite) N ensures that no player has an
  incentive to deviate
• Cost of punishment is extremely small as with
  time averaging pay-offs in a finite number of
  periods do not make a difference
• Average pay-oof to player j when I is punished is
  vj

11
Basic idea of SPE with minimax pay-offs discou
nted pay-offs

• Reward punishers, instead of punishing them if
  they dont punish
• Choose a vector in the interior of V such that
  for each i you can still give a higher pay-off.
• V needs to be of full dimension
• Play in three phases
• Initial cooperative phase
• Punishment phase where players minimax for N
  periods the deviator (as before) switch back to
  initial phase if this happens.
• If a player deviates in punishment phase start to
  punish that player
• What to do in case pay-offs can only be obtained
  with randomizations

12
Renegotiation proofness in repeated games

• Is SPE the best notion of a credible threat?
• Suppose you cooperate for some time in the PD and
  then someone defects, by chance. Should you go
  back immediately to always defect?
• Or should players renegotiate?
• It is in both players interest to revert back to
  the cooperative outcome
• In any equilibrium the equilibrium played must
  not be Pareto-dominated.
• Pareto-optimality as an assumption and the
  critique that is possible (risk dominance and
  Pareto-dominance)
• Deviations are accidents and unlikely to be
  repated? Bygones are bygones

13
Pareto perfection only applies in two-player games
A

• Two Nash equilibria in pure strategies (U,L,A)
  and (D,R,B)
• ULA is Pareto-efficient
• Natural candidate?
• Suppose players 1 and 2 expect matrix chooser to
  choose A. then they can renegotiate and gain by
  playing (D,R)

B
14
Definition of Pareto perfect equilibrium

• Fix stage game g and play it for T periods.
• Let P(T) the set of pay-offs of pure strategy SPE
  of G(T)
• R(t) is the set of strongly efficient points of
  P(t), i.e., this is the set of points such that
  there is not another pay-off point where no
  player is worse off and some player is better
  off.
• Set Q(1) P(1)
• For any t, let Q(t) be the set of pay-offs of
  pure strategy SPE that can be ebforced with
  continuation pay-offs in R(t-1)
• A SPE is Pareto perfect if for every possible
  history and in every time period t, the
  continuation pay-offs are in R(T-t)

15
Pareto perfection restricts threats

• Some efficient equilibria cannot be supported
  anymore under Pareto-perfection
• It restricts the set of threats, and thereby it
  is more difficult to keep players on the
  equilibrium path
• Example

16
Example Pareto-perfection

• Three pure strategies in G(1) with pay-offs
  (4,2), (2,4) and (3,3)
• In G(2) without discounting pay-off of 8 is
  possible. Unique element in R(2)
• Without restriction to Pareto perfection in G(3)
  pay-off of 13 possible
• With Pareto perfection in first period of G(3) no
  threat possible one has to play stage game
  equilibrium
• Equilibrium play alternates between odd and even
  periods under Pareto perfection

17
Exercises

• Fudenberg and Tirole 4.5
• Fudenberg and Tirole 5.1
• Consider the following normal form and the
  infinite repetition of it. What are the SPE of
  the infinite game? How does your result depend on
  d?