1. 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 єI is the finite set of players, Ai are finite action spaces
- gi:A→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 such that
- A mixed strategy for i is seq. of maps , where
is a probability distribution over Ai

3. Finite and infinite repeated games
Finite horizon games are solved using backward induction
Payoff functions for the infinite game G(δ)
, where (1-δ) is normalization factor
- for δ→1, we use time average criterion
The discount factor δ (<1) represents probability that the game may end at the end of any period
Thus, probability that the t-th stage will be played is δt

4. A useful result Theorem:
a. Consider a finitely repeated game. If α* is the Nash equil of the stage game, then the strategies “each player i plays α*i in every period” is a SPNE of the full game
b. If α* is unique Nash equil of the stage game, then the above strategies constitute the unique SPNE of the full game
Example: The finitely repeated Prisoner’s Dilemma
Confess
Not confess
0, 0
7, -2
-2, 7
5, 5 2 1

5. 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?

6. 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 πh/πl, with πl > πh.

7. 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

8. 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 50πh > 25πl.
b. Collusive case where 50πh < 25πl.

9. 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 unique Nash equilibrium is in mixed strategies
- in the multi price auction l is a dominant strategy, and the unique Nash equilibrium is (l, l)
Treasury prefers the single price auction!!

10. Infinitely repeated Prisoner’s 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 δ>2/7, then cooperation, (n, n) is sustainable!
Why the contrast with prediction from finitely repeated game?

11. Infinitely repeated Prisoner’s 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. δ→1?

12. The Folk Theorem for infinitely repeated games
Let player i’s reservation utility or minmax value be: .
This is the min value that his rivals can hold him to
Observation: Player i’s 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 єV, then there exists aєA, such that g(a)=v
The Folk Theorem:
For every feasible payoff vector v with vi > for all players i, there exists a <1 such that for all δє( , 1) there is a Nash equilibrium of G(δ) with payoffs v

13. 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 α* be a static equilibrium with payoffs e. Then for any vєV with vi > ei , for all players i, there is a such that for all δ> there is a subgame perfect equil of G(δ) with payoffs v.
Friedman’s conclusion is weaker than Folk theorem. Does subgame perfectness restrict set of equil payoffs?

14. Another Folk theorem Theorem (Aumann and Shapley 1976)
If players evaluate sequences of stage game utilities by the time average criterion, then for any vєV with vi > , 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