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A VERY BRIEF INTRODUCTION TO GAME THEORY In
game theory, we are usually looking for one or more equilibria (ideally only
one), which we regard as representing the likely outcome of a particular situation.
The principal criteria which an equilibrium is expected to satisfy are the
Nash equilibrium condition and 'subgame perfection'. Dominant
strategies We
can write the strategy of player i as si. By
strategy we mean a particular move or policy, e.g. “produce low output” (collude), or
“cooperate if and only if the other player did so the previous round” (trigger). We
can write the payoff to player 1 If u1(s1A,s2)
< u1(s1B, s2) for all
possible s2 and some s1B If a
strategy remains after iterative removal of all dominated strategies, it is a
rationalisable strategy. If
there is only one rationalisable strategy, it is the dominant strategy
(e.g. “defect” in the Prisoner's
Dilemma game). Nash
Equilibrium For
n players, (s1*,s2*,…,sn*) is
a Nash equilibrium (NE) if and only if: All
NEs consist of rationalisable strategies, but not all combinations of
rationalisable strategies are NEs. All
combinations of dominant strategies are NEs, but not all NEs consist of
dominant strategies. Mixed
strategy equilibrium A
mixed strategy is a probability distribution over strategies. I.e. a given
player is playing each of his possible strategies with some probability. For
example, firm A produces high output with 40% probability and low output with
60%, and firm B similarly mixes these strategies, but in the ratio 70:30. The
game may be one-shot, so the point isn’t necessarily that the players
alternate between strategies. In
this case the NE consists of optimal probability choices by each player given
the probability choices of other players. Once
you allow for mixed strategies then every (finite) game has at least one NE. Non-cooperative
game In
a cooperative game, the rules permit binding agreements prior to play. (Hence
collusion would be possible in a one-shot cooperative game.) In practice, we
are usually concerned with games in which this is not possible, i.e. a
non-coooperative game. Games
of complete information •
Players’ payoffs as functions of other players' moves are common knowledge. Static
and dynamic games In
a static (or ‘one-shot’) game, players move simultaneously and only once. In
a dynamic game, players either move alternately, or more than once, or both. Bertrand and Cournot models of
oligopoly competition are both static games, while the Stackelberg
model (firm B moves after firm A) is a dynamic game. An
equilibrium for a dynamic game must satisfy subgame perfection. Normal
and extensive forms A
game expressed in ‘normal form’ is in the form of a payoff matrix, as shown
below for the Prisoner's Dilemma game.
A
game expressed in ‘extensive form’ shows the ‘tree’ of the possible move
paths depending on what each player does at each stage. A dynamic game can
only be shown in extensive form.
Repeated
games The
same agents repeatedly playing a given one-shot game (“form game”) in
sequence is called a supergame. A supergame can consist of either a finite,
or an infinite, repetition of a form game. Or we could have a game with a
certain probability p of being repeated, i.e. a probability 1 – p
of breakdown. A strategy
in this context can be contingent on what other players have done in previous
moves. Subgame
perfection For
a dynamic game, some Nash equilibria are not acceptable as solutions because
one or more players will want to, and be able to, avoid those outcomes. The
subgame perfection criterion demands that, at each stage of the game, the
strategy followed is still optimal from that point on. Non-credible
threat A
'non-credible threat' is a strategy that one player is trying to use to
manipulate the behaviour of another (usually via the second move in a
sequential game), which forms part of a Nash equilibrium but one that is not
subgame perfect. The
strategy is one which is being claimed in some way by the threatening player
(e.g. by means of a signal that he is capable of using it) but which is not
credible: although the threatened player’s optimal response to the strategy is
to do what the threatening player wants, the former knows that by moving
first in a different way, the latter will adopt another strategy to generate
a different Nash equilibrium. In
a sense, there is no ‘credible threat’ — the term ‘threat’ implies that
player 1 will do something specifically designed to harm player 2 if
player 2 doesn’t comply, but such a threat would never be carried out in a
finite game with perfect information because such a move would not be optimal
for player 2. Player 2 will always ‘accommodate’ when it comes to it. Entry
deterrence 'Entry
deterrence' is an example of trying to manipulate a rival player's moves. In
this case, it involves an incumbent firm trying to prevent the entry of
potential rivals into a market. Successful
entry deterrence depends on avoiding the non-credible threat problem. If you
want to make things too difficult for a potential entrant to bother entering,
you have to do so in a way which binds you, i.e. you have to commit to a
particular strategy. This has to involve ex
ante (i.e. prior to the other player’s move) and irreversible action which
prima facie is suboptimal for player 1 (and therefore is said to be
‘strategic’, i.e. undertaken only for the purposes of affecting the other
player’s behaviour) but which ultimately pays because it succeeds in
deterring entry. Excess
(in the sense of surplus) capacity is not an effective way of
deterring entry; in fact it represents a non-credible threat. An incumbent
would never expand output in response to entry, he would always contract.
(Unless there is imperfect information, in which case he may try to convince
the other player he is irrational.) However, over-investment in capacity may succeed in
deterring entry. This is the Dixit* model in which the incumbent invests
irreversibly to expand the capacity at which he is able to produce at low marginal
cost, beyond what he would do left to himself. The point is that this results
in a post-entry equilibrium in which his output is higher than what it would
have been, and the entrant’s lower — so low that the latter can’t
cover its fixed cost. Moral
Hazard 'Moral
hazard' arises when player A wishes to contract with player B for the
performance of a variable task by B, the outcome of which will depend partly
on (i) B's effort and partly on (ii) random factors, and where it is
impossible to ascertain how much the outcome is due to (i) versus (ii). The
problem is that B does not have as much incentive to perform as would be
optimal. In the case of theft insurance, for example, the insured does not
have the ideal level of incentive to protect his property because the insurer
cannot monitor what he does, and he will therefore tend to under-protect it. There
is a connection between credible threats and moral hazard. To avoid moral
hazard, A wants B to believe there will be penalties for indulging in
'immoral' behaviour. However, the threat to penalise errant behaviour has to
be credible. Either the penalty has to be unavoidable, e.g. criminal legal
sanctions, or it has to be somehow in A's interests to apply it. The problem
is that the application of a punishment is not usually intrinsically
beneficial for the punisher. One possible way out is through reputation: if
A's reputation for truth-telling and toughness is valuable to A, then A
announcing publicly that a penalty will be imposed could lead to a cost for A
if he then fails to implement. In this way, the threat to punish would become
credible. Applying
this to the Bank of England, a threat not
to bail out banks in trouble except in very limited circumstances is at
risk of not being credible and therefore of not being effectual, unless
reneging on the threat can be regarded as somehow costly for the Bank.
However, it is not clear how the Bank, or any of its agents, could suffer
from the failure to penalise an errant lender. Possibly when the Bank was
still relatively controlled by the government (pre-1997), the desire of the
ruling party to be re-elected could have provided such an incentive. When
there is imperfect information about whether the failure to carry out
a threat is costly for the threatener, it is possible for the threat to be
credible by exploiting uncertainty. However, once a player has reneged on his
threat without obvious negative repercussions, the possibility of future
credible threats is more or less eliminated. Fabian
Tassano * Dixit, A. (1980) 'The Role of Investment in Entry Deterrence', Economic
Journal 90, 95-106. |
