ECON 3210/4210 Decisions, Markets and Incentives
Lecture notes 21.09.05
Nils-Henrik von der Fehr
STATIC GAMES OF COMPLETE INFORMATION
Introduction
Multi-person decision problems
Strategic interaction
Examples
„
oligopolistic interaction
„
bargaining and negotiations
„
principal-agent theory
„
political economy
In this part we consider games of the following form:
„
first players simultaneously choose actions (simultaneous moves);
„
then players receive payoffs that depend on the combination of actions
chosen;
„
each player’s payoff function, as well as the rules of the game, is common
knowledge (complete information)
Topics
„
how to describe a game (representation)
„
how to solve a game (equilibrium concepts)
An example
Consider two firms who each may choose to introduce a customer loyalty
programme, such as a frequent flyer programme common in passenger air
transport. The programme itself is costly, but if it succeeds in attracting and
maintaining a larger market share the increase in sales revenues may more
than outweigh the additional costs.
Let payoffs to the two firms be given as follows, where the first number gives
the payoff of the Firm 1 and the second number gives the payoff of Firm 2:
Firm 2
No loyalty programme
Loyalty programme
No loyalty programme
0,0
-2,1
Loyalty programme
1,-2
-1,-1
Firm 1
The game is commonly referred to as a Prisoners’ Dilemma type game. The
original story has two suspects on prison who may either confess or not. If
neither confesses, they are convicted for minor offences only. If both confess,
both will be sentenced to jail. If one confesses, but the other does not, the
confessor is released while the other gets an additional sentence for
obstructing justice.
Note that only the outcome (-1,-1) is reasonable (i.e., in which both firms
choose to introduce a loyalty programme or both prisoners confess).
Normal-form representation
The normal-form representation of a game specifies:
1)
the players in the game;
2)
the strategies available to each player;
3)
the payoff received by each player for each combination of strategies that
could be chosen by players.
We make the following definitions:
„
there are n players, with an arbitrary player called player i, i = 1,2,...,n;
„
Si denotes the set of strategies available to player i (strategy space), while
si denotes an arbitrary member of this set (so si ∈ Si );
„
( s1,..., sn ) is a combination of strategies for all players (a strategy profile);
2
„
ui ( s1,..., sn ) denotes the payoff to player i for the strategy combination
( s1,..., sn ) .
The normal-form representation of an n-player game specifies the players’
strategy spaces S1,..., Sn and their payoff functions u1,..., un . Such a game is
denoted G = {S1,..., Sn ; u1,..., un } .
An interpretation of the simultaneous-move structure is that each player
chooses his or her action without knowledge of the others’ choices.
Iterated elimination of dominated strategies
Note that in the above example, introducing a consumer loyalty programme is
preferable whatever the competitor chooses to do. In other words, the strategy
‘no loyalty programme’ is dominated by the strategy ‘loyalty programme’.
More generally, we have the following definition: In the normal-form game
G = {S1,..., Sn ; u1,..., un } , the strategy si′ ∈ Si is strictly dominated by the
strategy si′′ ∈ Si if, for each ( s1,..., si −1, si +1,..., sn ) , sk ∈ Sk ,
ui ( s1,..., si −1, si′, si +1,..., sn ) < ui ( s1,..., si −1, si′′, si +1,..., sn ) .
In other words, the strategy si′′ yields a higher payoff than si′ whatever the
strategy choices of the opponents.
Rational players do not play strictly dominated strategies! Note that this leads
to a unique equilibrium in the Prisoners’ Dilemma game considered above.
More generally, when, for all players, all strategies except one is dominated by
one particular strategy we have what is called a dominant-strategy equilibrium.
In the Prisoners’ Dilemma game, the strategy combination (loyalty
programme, loyalty programme) is a dominant-strategy equilibrium.
A weaker solution concept is based on iterated elimination of strictly
dominated strategies. Consider the following example:
Player 2
Left
Middle
Right
Up
1,0
1,2
0,1
Down
0,3
0,1
2,0
Player 1
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Note that, for Player 2, the strategy Right is strictly dominated by the strategy
Middle. Consequently, Player 1, knowing the preferences of Player 2, should
consider it impossible that Player 2 will play Right and hence may eliminate
this strategy from the strategy space of Player 2. This leads to the following
game:
Player 2
Left
Middle
Up
1,0
1,2
Down
0,3
0,1
Player 1
Now, given that Player 1 can rule out that Player 2 will ever play Right, for him
the strategy Down is strictly dominated by the strategy Up. Consequently,
Player 2, knowing that Player 1 will reason that Player 2 will never play Right,
should consider it impossible that Player 1 will play Down and hence may
eliminate this strategy from the strategy space. This leads to:
Player 2
Player 1
Up
Left
Middle
1,0
1,2
Now, for Player 2, the strategy Left is strictly dominated by the strategy
Middle. We conclude that, in this case, iterated elimination of strictly
dominated strategies leads to a unique solution of the game, namely the
strategy combination (Up, Middle).
Although the process of iterated elimination of dominated strategies is based
on the appealing idea that rational players do not play strictly dominated
strategies, it has two drawbacks:
„
it requires ever further assumptions about what players know about each
other’s rationality. If the process is to be applied for an arbitrary number of
steps, we need to assume that it is common knowledge that players are
rational; that is, all players are rational, all players know that all players are
rational, all players know that all players know that all players are rational,
and so on ;
„
it often produces very imprecise prediction of play of the game; i.e., the
process often stops before all but one strategy for each player are
eliminated.
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To illustrate the last point, consider the following example. A new type of
consumer product is about to be introduced (relevant examples include music
machines and 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 sales for itself.
The strategic choice involves whether to choose ones own technology or that
of its competitor. Payoffs are as follows:
Firm 2
Own technology
Competitor’s
technology
Own technology
0,0
2,1
Competitor’s technology
1,2
0,0
Firm 1
In this game, no strategy is strictly dominated by another. Hence, the process
of elimination of strictly dominated strategies has no bite and hence provides
no solution to how the players will play this game.
Nash Equilibrium
Motivation: in order to form an equilibrium, a strategy combination should be
such that
„
each player consider his or her strategy a best response to what
everybody else is expected to play (no regrets), or,
„
play should be strategically stable or self-enforcing.
Definition: in the n-player normal-form game G = {S1,..., Sn ; u1,..., un } the
(
)
strategies s1* ,..., sn* constitute a Nash equilibrium if, for each player i,
(
)
(
)
ui s1* ,..., si*−1, si* , si*+1,..., sn* ≥ ui s1* ,..., si*−1, si , si*+1,..., sn* , si ∈ Si
In other words, the strategy si* is a best response to the strategies
( s ,..., s
*
1
*
i −1
)
, si*+1,..., sn* of the n-1 other players and solves
(
)
max si ∈Si ui s1* ,..., si*−1, si , si*+1,..., sn* .
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It follows that, by definition, for a strategy combination ( s1,..., sn ) that does not
constitute a Nash equilibrium, at least one player will have an incentive to
deviate to another strategy.
Note also that players’ strategies in a Nash equilibrium always survive iterated
elimination of strictly dominated strategies, but the converse is not true.
Hence, Nash equilibrium is a stronger solution concept.
In the Prisoner’s Dilemma game, the dominant-strategy equilibrium is also a
Nash equilibrium. We can check this by first demonstrating that for all other
strategy combinations at least one player would want to deviate. Second, at
equilibrium, playing the equilibrium strategy is a best response. Alternatively,
we may apply the result that if iterated elimination of strictly dominated
strategies eliminate all but one strategy combination, this combination is the
unique Nash equilibrium.
Consider next the technology-choice example introduced immediately above.
This is an example of a so-called coordination game and is in this particular
form often referred to as a Battle of the Sexes type game. The original story
has a man and a woman having to choose (without communicating!) whether
to go to a boxing match or to the opera. They would both like to go together,
but the man prefers the boxing match and the woman the opera. In this game
there are two Nash equilibria: in the technology-choice interpretation these are
(own technology, competitor’s technology) and (competitor’s technology, own
technology) (in the traditional interpretation they are (boxing match, boxing
match) and (opera, opera)).
More generally, the Nash solution concept may produce one, many or no
pure-strategy equilibria (if we allow for mixed-strategy equilibria, in which
players randomise between pure strategies, there always exists an equilibrium
in these kinds of games).
In the case of many Nash equilibria, it may sometimes be possible to invoke
additional criteria for selecting a particular equilibrium. Consider the following
variant of the technology-choice example:
Firm 2
Own technology
Competitor’s
technology
Own technology
0,0
3,2
Competitor’s technology
1,2
0,0
Firm 1
The interpretation may be the following: Firm 1’s technology is superior to
Firm 2’s and hence payoffs are higher when this technology becomes the
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standard (in fact, here (own technology, competitor’s technology) Pareto
dominates (competitor’s technology, own technology) from the point of view of
the two firms). It may then seem reasonable to choose (own technology,
competitor’s technology) as the solution to the game.
In other cases, there is no natural ‘focal point’ and hence the Nash equilibrium
does not provide a solution to the game.
Application: Tragedy of the Commons
There are n fishermen. Fisherman i, i = 1,2,...,n, must choose his fishing
capacity (size and type of boat, crew etc.), which, for simplicity, we think of as
a one-dimensional continuous variable si ∈ Si [0, ∞ ) . The unit cost of capacity
is constant and equal to c. The value of fisherman i’s catch, when the total
n
capacity of the fleet is s = ∑ i =1 si , is v ( s ) si . It follows that the payoff to
fisherman i, when the strategy combination is ( s1,..., sn ) , is
⎛
n
⎞
π i ( s1,..., si ) = v ⎜ ∑ si ⎟ si − csi .
⎝ i =1
⎠
We assume that the function v – value of catch per capacity unit – is
decreasing and convex; that is, v ′ < 0 and v ′′ < 0 (value of catch may be
decreasing because a larger fishing fleet – and hence competition among
fishermen – increases the effort – i.e. hours at sea – required to obtain a
certain volume of catch; also, a higher catch may depress prices on fish). We
further assume that V ( s ) = v ( s ) s – the value of total catch – is increasing
everywhere; that is, V ′ ( s ) > 0 . The figure below provides an example:
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(
)
At the Nash equilibrium s1* ,...sn* , Fisherman i has chosen capacity so as to
maximise payoff, given the choices of all other fishermen. The first-order
condition for the optimality of Fisherman i’s choice is
⎛ n ⎞
⎛ n ⎞
v ⎜ ∑ s *j ⎟ + v ′ ⎜ ∑ s *j ⎟ si* = c .
⎝ j =1 ⎠
⎝ j =1 ⎠
On the left-hand side is the gross gain from marginally increasing capacity.
This consists of the extra catch provided by the marginal capacity unit ( v )
less the reduction in catch on all inframarginal units as a result of an overall
increase in capacity ( v ′ ⋅ si ). This marginal gain should equal marginal cost
(c).
Summing over all i = 1,2,...,n, and dividing by n, gives
(*)
v (s *) +
1
v ′ (s * ) s* = c .
n
We may contrast the outcome of the game (the decentralised solution) with
the total capacity that maximises the net value of total catch, i.e. v ( s ) s − cs .
The first-order condition for this problem is
(**)
v (s ) + v ′ (s ) s = c .
The solution is illustrated by the point s’ in the figure above.
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Comparing (*) and (**), we find that s * > s ' , that is, the Nash equilibrium
results in a total capacity that exceeds the capacity that maximises the value
of total catch. The intuition is that each fisherman only considers the reduction
in value on unit catch on his own capacity and not that of all other fishermen.
In other words, increasing individual capacity involves a negative externality
on all others.
A formal proof of the above result may proceed as follows. Assume, for
contradiction, that s * ≤ s ' . Then
c = v (s ') + v ′ (s ') s '
≤ v (s * ) + v ′ (s * ) s *
< v (s * ) +
1
v ′ (s *) s * .
n
The equality follows from (**); the first inequality follows from the assumptions
that s * ≤ s ' and v is decreasing and convex, which implies that V ′ is
decreasing also; and the last inequality follows from the assumption that n > 1
and v ′ < 0 . This result, however, contradicts (*). It follows that the assumption
s * ≤ s ' must be wrong; that is, s * > s ' . QED.
The model illustrates the more general idea that, if individuals respond to
private incentives only, public (i.e. freely accessible) resources will be overutilised (and public goods under-provided).
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