Dynamic Games of Complete Information

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Dynamic Games of Complete Information

• .

2
Repeated games

• Best understood class of dynamic games
• Past play cannot influence feasible actions or
  payoff functions
• Building block is called stage game
• - i ?I is the finite set of players, Ai are
  finite action spaces
• - giA?R, are payoff functions where A
• - players move simultaneously
• - ht is history before period t, ht(a0, a1,
  at-1), and Ht(A)t is space of all period-t
  histories at
• - A pure strategy for i is seq. of maps
  
• - A mixed strategy for i is seq. of maps
  , where
• is a probability distribution over
  Ai

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Finite and infinite repeated games

• Finite horizon games are solved using backward
  induction
• Payoff functions for the infinite game G(d)
• - , where (1-d) is
  normalization factor
• - for d?1, we use time average criterion
• The discount factor d (lt1) represents probability
  that the game may end at the end of any period
• Thus, probability that the t-th stage will be
  played is dt

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A useful result

• Theorem
• a. Consider a finitely repeated game. If a is
  the Nash equil of the stage game, then the
  strategies each player i plays ai in every
  period is a SPNE of the full game
• b. If a is unique Nash equil of the stage game,
  then the above strategies constitute the unique
  SPNE of the full game
• Example The finitely repeated Prisoners Dilemma

Confess Not confess
Confess 0, 0 7, -2
Not confess -2, 7 5, 5
2
1
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An example Treasury bills auction

• US Treasury Dept periodically sells securities
• Sold by auction to large financial institutions
• Auctions held on a regular basis
• There are two kinds
• - single price auctions (one price for all
  buyers)
• - multi price auctions (different prices)
• For any one kind of security this is repeated
  game
• Which of the two forms should Treasury use?

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Treasury bills auction

• Simplifying assumptions
• 1. two financial institutions
• 2. quantity of bills, 100, fixed across auctions
• 3. buyers can offer two prices two quantities
• -prices can be high (h) or low (l)
• -quantity can be 50 or 75
• -profit per security with high / low price are
  ph/pl, with pl gt ph.

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Treasury bills auction

• If both firms offer a high price, then market
  price is high and total demand is 100
• If both firms offer a low price, then market
  price is low
• If one wants to buy at h and other at l, then
• - in single price auction price is l
• - in multi price auction one pays h the other,
  l
• - high bidder gets his full qty, rest goes to
  rival
• If price bids are the same, allocation is
  proportionate to qty demanded

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Treasury bills auction

• Note At any price it is always better to ask for
  a larger quantity
• Therefore we can look at the reduced games
• Consider two cases
• a. Competitive case where 50ph gt 25pl.
• b. Collusive case where 50ph lt 25pl.

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Treasury bills auction

• Competitive case
• - in the single price auction h is a dominant
  strategy, and the unique Nash equilibrium is (h,
  h)
• - in the multi price auction both (h, h) and (l,
  l) can be Nash equilibria
• Collusive case
• - in the single price auction the Nash
  equilibria are
• (h, l) and (l, h). There is also a mixed
  strategy
• - in the multi price auction l is a dominant
  strategy, and the unique Nash equilibrium is (l,
  l)
• Treasury prefers the single price auction!!

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Infinitely repeated Prisoners Dilemma

• Consider the grim trigger strategy
• a. Start by playing (n, n) and continue playing
  it as long as no one confesses
• b. If anyone confesses, play (c, c) from then
  on
• This is a SPNE
• If dgt2/7, then cooperation, (n, n) is
  sustainable!
• Why the contrast with prediction from finitely
  repeated game?

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Infinitely repeated Prisoners Dilemma

• Two important points
• 1. Grim punishments may achieve other behaviors
• 2. Cooperative behavior is achievable with less
  severe punishments
• Example of point 1
• -Start with (n, c). Play (n, c) at even numbered
  periods and (c, n) at odd ones. If there is
  deviation, play (c, c) from then on.
• - Show that above is credible
• Example of point 2
• - A Forgiving trigger strategy says, play (n, n)
  and if there is deviation play (c, c) for T
  periods. Revert to (n, n)
• - Is this credible?
• - What happens when future is very important,
  i.e. d?1?

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The Folk Theorem for infinitely repeated games

• Let player is reservation utility or minmax
  value be
• .
• This is the min value that his rivals can hold
  him to
• Observation Player is payoff is at least in
  any Nash equilibrium of the stage game, and
  repeated game, regardless of the discount factor
• Let V be set of feasible payoffs, i.e. if v ?V,
  then there exists a?A, such that g(a)v
• The Folk Theorem
• For every feasible payoff vector v with vi gt for
  all players i, there exists a lt1 such that for
  all d?( , 1) there is a Nash equilibrium of
  G(d) with payoffs v

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Nash-threats Folk theorem

• Strategy used in proof of Folk thm Let g(a)v.
  Play ai in all periods until there is a
  deviation. After a deviation by i (say), all
  players -i play the minmax profile m-ii which
  gives i a payoff
• The above strategies are not subgame perfect
• Theorem (Friedman 1971)
• Let a be a static equilibrium with payoffs e.
  Then for any v?V with vi gt ei , for all players
  i, there is a such that for all dgt there
  is a subgame perfect equil of G(d) with payoffs
  v.
• Friedmans conclusion is weaker than Folk
  theorem. Does subgame perfectness restrict set of
  equil payoffs?

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Another Folk theorem

• Theorem (Aumann and Shapley 1976)
• If players evaluate sequences of stage game
  utilities by the time average criterion, then for
  any v?V with vi gt , there is a subgame perfect
  equilibrium with payoffs v
• Idea behind proof
• a. Use strategy Play strategy that gives v as
  long as there are no deviations. If i deviates
  play minmax profile m-ii which for N periods,
  where,
• b. With the time average criterion, minmaxing a
  deviator is not costly