Advanced Microeconomics (EC30025)
Topic One: Introduction to Game Theory (ii)
Advanced Microeconomics (ES30025)
Topic One: Introduction to Game Theory (ii)
Outline:
1.
1.
2.
Dynamic Games
Solving Dynamic Games: Sub-Game Perfection
Dynamic Games
Let us reconsider games in which there is a sequence of moves. Recall Game One:
1
R
L
2
L
(2,0)
2
R
(2,-1)
L
(1,0)
R
(3,1)
Figure 1: Game One
Te absence of any information set around Player 2’s nodes indicate that the game is
sequential with each node in the game tree representing a distinct information set such
that Player 2 knows at which of his two nodes he is situated. The game must, therefore,
be a sequential one in which Player 1 makes the first move and then Player 2 follows.
Thus Player 2 knows whether Player 1 has moved left or right before he (Player 2)
makes his move.
In a dynamic game at least one player chooses his strategy after finding out
(something about) what the other player has chosen. In such a game, a strategy is
defined a complete specification of the moves a player may make in the entire course of
the game. Player 1 still has just two strategies (or actions), namely to move left or right.
Player 2’s strategies are, however, a little more complex, and do not correspond with
his actions. Since a strategy must specify in advance how a player will move in each
possible contingency, Player 2 has four possible strategies open to him: to always move
left; always move right; to follow Player 1; or to do the opposite of Player 1. Thus:
1
Advanced Microeconomics (EC30025)
Topic One: Introduction to Game Theory (ii)
( )
= ( s ,s ,s ,s ) = {( L, L ) , ( R, R ) , ( L, R ) , ( R, L )}
s1 = s11 ,s12 = ( L, R )
s2
1
2
2
2
3
2
(7)
4
2
where s12 = (L, L ) is read: If s11 = L then chose L, and if s12 = L then chose L (i.e.
always chose L). The normal form representation of the game is:
Player 2
Player 1
s11 = L
s12 = R
s21
(L, L)
2, 0
s22
s23
s12
(R, R)
2,-1
(L, R)
2, 0
(R, L)
2, -1
1, 0
3, 1
3, 1
1, 0
Table 1: Game One
2.
Solving Dynamic Games: Sub-Game Perfection
We can solve dynamic games in either the normal or extensive form.
The Normal Form
Reconsidering Game One, it is apparent that there are three Nash-equilibrium strategy
pairs, namely:
{
Nash Equlibria: s* = ⎡⎣ s11 = L,s21 = ( L, L ) ⎤⎦ ; ⎡⎣ s12 = R,s22 = ( R, R ) ⎤⎦ ; ⎡⎣ s12 = R,s23 = ( L, R ) ⎤⎦
}
To check this - consider the Normal Form representation of the game reproduced in
Table 3 following:
s22
s23
s12
s =L
s21
(L, L)
2, 0
(R, R)
2,-1
(L, R)
2, 0
(R, L)
2, -1
s =R
1, 0
3, 1
3, 1
1, 0
Player 2
Player 1
1
1
2
1
Table 2: Game One – Multiple PSNE
In words:
s12 is Player 2’s ‘best response’ to s11 , and s11 is Player 1’s ‘best response’ to s12 .
s22 is Player 2’s ‘best response’ to s12 , and s12 is Player 1’s ‘best response’ to s22 .
s23 is Player 2’s ‘best response’ to s12 , and s12 is Player 1’s ‘best response’ to s23 .
Note that the ‘mutual best response’ property is possessed by no other strategy pair.
How do we reconcile the three PSNE above to the apparently unique PSNE we
found in the extensive form via. backward induction? The point is, the equilibrium we
found in the extensive form is a special type of PSNE. Consider the three PSNE:
Consider first ⎡⎣ s12 = R, s22 = (R, R )⎤⎦ . We can interpret this strategy pair as
embodying a threat on the part of Player 2. Player 2 is effectively saying to Player 1 that
2
Advanced Microeconomics (EC30025)
Topic One: Introduction to Game Theory (ii)
he (Player 2) will always play R. The incentive for doing this is to coerce Player 1 into
playing R as well. Player 2’s most preferred outcome is (3, 1), and by threatening to
play R always he hopes that Player 1 will be encouraged to play R as well since to
choose L would imply the payoff (2, -1). The question arises as to the credibility of this
threat. If Player 1 chooses L then Player 2 will have to do the best he can given this
fact. But clearly this would require Player 2 playing L rather than R, yielding the payoff
(2, 0). Thus, the threat by Player 2 to play R always is not credible because he has no
incentive to carry it out when he is placed in the position to do so. The strategy
s22 = (R, R ) is therefore incredible.
A similar argument would also reject ⎡⎣ s11 = L, s12 = (L, L )⎤⎦ as a reasonable
strategy pair since it requires Player 2 to chose L if Player 1 chooses R, when R is
clearly better for Player 2. Thus although ⎡⎣ s11 = R, s22 = (R, R )⎤⎦ would result in the same
outcome as ⎡⎣ s12 = R, s23 = (L, R )⎤⎦ (since it prescribes the same choice for Player 2 in the
event that Player 1 chooses R which she does in this strategy pair), if we require that a
strategy specifies rational behaviour at each and every possible point in the game, then
⎡⎣ s11 = R, s22 = (R, R )⎤⎦ should also be ruled out.
Consider finally the Nash-equilibrium strategy pairs ⎡⎣ s12 = R, s23 = (L, R )⎤⎦ . This
has Player 2 behaving rationally in all contingencies and is thus the only Nashequilibrium strategy pair that could not be rejected as unreasonable.
The above argument seems to suggest that of the three Nash-equilibrium pairs,
only ⎡⎣ s12 = R, s23 = (L, R )⎤⎦ is a reasonable prediction of the game, and for this reason is
known as a Sub-Game Perfect Equilibrium (SGPE).
s12
s22
s23
s12
s =L
(L, L)
2, 0
(R, R)
2,-1
(L, R)
2, 0
(R, L)
2, -1
s =R
1, 0
3, 1
3, 1
1, 0
Player 2
Player 1
1
1
2
1
Table 4: Game One – Sub-Game Perfect Equilibrium
The Extensive Form
Sub-game perfection can perhaps be most readily derived through the extensive form.
Consider Figure 2 following. We can use a process of backward induction known as
Kuhn’s Algorithm (or ‘folding back the tree’). Beginning at the end - i.e., at the last time
a decision is made. In the present game, Player 2 makes his decision last.
3
Advanced Microeconomics (EC30025)
Topic One: Introduction to Game Theory (ii)
1
R
L
2
L
2
R
(2,0)
L
(2,-1)
R
(1,0)
(3,1)
Figure 2: Game One (Again)
Player 2 makes his decision as follows: If he observes Player 1 has played L, he will
also play L, since to do otherwise (i.e. to play R) yields a payoff of –1 < 0. If Player 2
observes Player 1 has played R, he will also play R, to do otherwise (i.e. to play L)
yields a payoff of 0 <1. We can therefore disregard two branches of the tree vis. Player
2 playing R if Player 1 plays L, and Player 2 playing L if Player 1 plays R.
1
R
L
2
L
(2,0)
2
R
L
(2,-1)
(1,0)
Figure 3
4
R
(3,1)
Advanced Microeconomics (EC30025)
Topic One: Introduction to Game Theory (ii)
The critical point is that Player 1 realizes that this is how Player 2 will behave when he
(Player 1) makes his move. Moving up the extensive form, Player 1 knows that if he
plays L then Player 2 will also play L, which will yield him (Player 1) a payoff of 2;
whereas if he plays R, then Player 2 will also play R, thereby yielding him (Player 1) a
payoff of 3.
1
R
L
2
L
(2,0)
2
R
L
(2,-1)
R
(1,0)
(3,1)
Figure 4
Player 1 will therefore choose to play R.
The concept of sub-game perfection was introduced by Selten and generalizes the
notion of reasonableness underlying this example. Its effect is to narrow down the
equilibria of a game to a subset of the Nash equilibria, and for this reason is often
referred to as a ‘refinement’ of Nash equilibria. To locate a sub-game perfect equilibria
in a game with a sequence of moves, take a decision node at any point in the game
(including the first node) and identify the sequence of moves after that point as a game
itself: i.e. a sub-game of the original game.1 Formally:
A sub-game perfect equilibrium arises when the players’ strategies for a
particular game induce a Nash equilibrium in every sub-game of that
game.
Note that since a game is a sub-game of itself, this implies that:
All sub-game perfect equilibria are Nash equilibria, but not all Nash
equilibria are sub game perfect.
The requirement of Nash equilibrium behaviour in sub-games ensures the credibility of
threats: a threat which corresponds to playing a Nash equilibrium strategy in a sub1
Thus in Figure 1 there are three sub-games: the two games beginning at each of Player 2’s decision
nodes, which consist of only one move by Player 2, and the original game itself.
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Advanced Microeconomics (EC30025)
Topic One: Introduction to Game Theory (ii)
game beginning at a particular node within the game is credible, whilst an action that is
not consistent with Nash equilibrium in the sub-game will not be.
Sub-game perfection therefore requires that a reasonable solution to a game
cannot involve players believing and acting upon incredible threats or promises. More
formally, a sub-game perfect equilibrium requires that the predicted solution to a game
be a Nash equilibrium in every sub-game of the overall game. A sub-game is defined as
a smaller part of the whole game, starting from any particular node and continuing to
end of the entire game, with the qualification that no information set is subdivided. A
sub-game is therefore a game in its own right that may (or may not!) be played in the
future, and as such is a relevant part of the overall game. By requiring that a solution to
a dynamic game must be a Nash equilibrium in every sub-game amounts to saying that
each player must act in his own self-interest in every period of the game. This means
that incredible threats (or promises) will not be acted upon.
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