ECON 3210/4210 Decisions, Markets and Incentives
Lecture notes 10.10.05
Nils-Henrik von der Fehr
DYNAMIC GAMES OF COMPLETE
INFORMATION
Introduction
Static versus dynamic games
Sequential move order
„
some players act only after having observed others’ choice
„
early movers may be able to affect play of later movers
Extensive Form vs. Normal Form representations
Refinements of Nash Equilibrium
„
backward induction
„
Subgame Perfect Equilibrium
An example
Consider again the following example. A new type of consumer product is
about to be introduced (relevant examples include music machines and
computer game consoles). There are two competing technologies, controlled
by different firms. Both firms would like there to be one standard, as this would
increase total sales. However, each firm would like its own technology to
become the standard, as this would mean higher revenues for itself (from own
sales as well as from licensing of technology to the competitor). The strategic
choice involves whether to choose ones own technology or that of its
competitor. Payoffs are as follows:
Firm 2
Technology 1
Technology 2
Technology 1
2,1
0,0
Technology 2
0,0
1,2
Firm 1
In this game there are two Nash equilibria: both firms choosing Technology 1
and both firms choosing Technology 2.
Consider now the Extensive Form representation of this game, in which we
assume that Firm 1 makes its choice before Firm 2.
Figure: game tree
Game tree
„
decision node
„
payoffs given at end nodes
Again, two Nash equilibria, but is the Nash equilibrium in which both firms
choose Technology 2 reasonable?
Subgame Perfection
Backward induction in the above example:
„
start at nodes at which Firm 2 makes decision and derive optimal choices
in these subgames;
„
go back to node at which Firm 1 makes decisions and consider Firm 1’s
choice given that Firm 2 is expected to choose optimally.
Subgame
„
part of a larger game
Definition of Subgame Perfect Equilibrium: Strategies constitute a Subgame
Perfect Equilibrium if they constitute a Nash equilibrium of all subgames of the
game (including the game itself).
Commitment
„
Classic example: Cesar at Rubicon.
2
„
cf. Nobel-Price winner Thomas C. Schelling, “The Strategy of Conflict”
Repeated Games
Repetition of one-shot (or stage) games
„
supergame
„
finite versus infinite repetitions
Actions versus strategies
„
action is the choice at any particular stage
„
strategy defines a plan of actions
„
contingent strategies depend on what happened earlier in the game
Finite repetitions
Example: the repeated Prisoners’ Dilemma
Player 2
L
R
U
0,0
-2,1
D
1,-2
-1,-1
Player 1
Consider first the last stage of the game: it has a unique equilibrium.
Consequently, at the second-to-last stage, players should realise that play in
the last stage will not be influenced by what they do at the current stage and
hence should play the one-shot equilibrium here also. Similar backwardinduction reasoning establishes that the (subgame perfect) equilibrium is
unique.
This example hints at a more general result: when the stage game has a
unique Nash equilibrium then, for any finite number of repetitions, the
repeated game has a unique Subgame Perfect equilibrium in which the Nash
equilibrium is played in every stage.
3
However, when the stage game has multiple equilibria, the repeated game
may have Subgame Perfect equilibria that do not correspond to equilibria of
the one-shot game.
Example: Cabral
Player 2
Player 1
L
C
R
U
5,5
3,6
0,0
M
6,3
4,4
0,0
D
0,0
0,0
1,1
There are two Nash equilibria in the one-shot game: (M,C) and (D,R).
However, the outcome (5,5) may be sustained as an equilibrium if the game is
repeated twice (or more), by the following strategies:
„
Player 1: at stage 1, play U. At the second stage, play M if (U,L) was
played in the first stage; otherwise, play D.
„
Player 2: at stage 1, play L. At the second stage, play C if (U,L) was
played in the firs stage; otherwise, play R.
Clearly, at the second stage, in both subgames strategies constitute a Nash
equilibrium. Moreover, payoffs from equilibrium play is 9 (= 5+4) for both
players, while the maximum achievable from any other strategy is 7 (6+1).
Hence, strategies constitute a Nash equilibrium of the overall game also.
Note, the threat of triggering a ‘bad’ equilibrium disciplines players. In other
words, threats and promises may affect future behaviour.
But, are they credible in the above example? When reaching the second
stage, would it not be reasonable to play (M,C) – since this payoff dominates
(D,R) – whatever happened at the first-stage; in particular, should not the
players ‘renegotiate’ after a first-stage deviation? If they do, however, (U,L)
cannot be sustained as an equilibrium in the first stage.
One implication is that threats and promises are credible only if they are not
based on payoff-dominated outcomes (see Gibbons, 87-88).
How would payoff-maximising equilibrium strategies look in a corresponding
game with T > 2 repetitions?
4
Infinite repetitions
Example: the Prisoners’ Dilemma game.
Suppose:
„
the outcome of stage t-1 is observed before stage t begins;
„
players discount payoffs with factor δ ∈ [0,1] .
Denoting the stage payoff to player i by π it and the current period by 0, the
total, discounted profits become
∞
Π i = ∑ δ t π it .
t =0
Consider the following set of strategies:
„
Player 1: at the first stage, play U. Continue to play U so long as (U,L) has
been played in all previous stages; otherwise, play D;
„
Player 2: at the first stage, play L. Continue to play L so long as (U,L) has
been played in all previous stages; otherwise, play D.
Comparison of payoffs. Payoffs along the equilibrium path – i.e. always play of
(U,L) – are 0. Payoff to Player 1 from deviating to D in the current round is
∞
Π1 = 1 + ∑ δ t [ −1] = 1 −
t =1
δ
1− δ
=
1 − 2δ
.
1− δ
This is the same that Player 2 would obtain when playing R in the current
round.
The critical discount factor such as to make (U,L) a Subgame Perfect
Equilibrium is δ = 0.5 ; that is, (U,L) can be sustained as an equilibrium if and
only if δ ≥ δ = 0.5 .
Folk theorem: When players are sufficiently patient, any feasible combination
of payoffs that yields to each player as least as much as in some Nash
equilibrium of the one-shot game, may be sustained as the average payoff of
the infinitely repeated game.
Conclusion: in infinitely repeated games there may be equilibria that do not
correspond to equilibria of the one-shot game even if the one-shot game has
a unique equilibrium.
5
Interpretation of discount factor: suppose the interest rate is r and that with
probability p the game stops after the current period. Then the discount factor
may be written:
δ=
1− p
1+ r
In other words, infinite repetitions may be associated with the case in which
there is no fixed end date.
Perfect and imperfect information
Any game – whether static or dynamic – may be represented in both Normal
Form and Extensive Form.
Consider the following Normal-Form Representation:
Player 2
L
R
U
0,2
-2,1
D
1,-2
-1,-1
Player 1
The game has a unique Nash equilibrium (D,R) (this solution may also be
found by iterated elimination of dominated strategies).
We may formulate the simultaneous-move game in extensive form, by using
the concept of information sets.
An information set for a player is a collection of decision nodes satisfying (i)
the player has the move at every node in the information set, and (ii) when
play of the game reaches a node in the information set, the player with the
move does not know which node in the information set has been reached.
In the game tree, we may indicate that a collection of decision nodes
constitutes an information set by connecting the nodes by a line (alternatively,
drawing a circle around these nodes).
Figure: game tree
Consider then the dynamic version of the game in which Player 1 moves first.
Figure: game tree
6
In this game there is a unique Subgame Perfect Equilibrium strategy profile
(U,(L,R)), where the formulation (L,R) is used to indicate that Player 2’s
strategy is contingent; that is, Player 2 chooses L if Player 1 chooses U and R
if Player 1 chooses D. The strategy profile (D,R) is a Nash Equilibrium, but not
a Subgame Perfect Equilibrium.
This game captures the idea that commitment is valuable: by choosing U first,
Player 1 forces Player 2 to play the strategy L.
Perfect information may defined to mean that at each move in the game the
player with the move knows the full history of the play of the game thus far.
Equivalently, perfect information means that every information set is a
singleton. Imperfect information then means that there is at least one nonsingleton information set in the game.
A dynamic game of complete but imperfect information can be represented in
extensive form by using non-singleton information sets to indicate what each
player knows (and does not know) when he or she has to the move.
7