12. A Characterisation of 2 × 2 Symmetric Games
A symmetric 2 × 2 matrix game can be described in the following
form
A
B
A
(a, a)
(c, b)
B
(b, c)
(d, d)
Since it suffices to give the payoff of the first player, this can be
simplified to
A
B
A
a
c
B
b
d
It will be assumed that a, b, c, d are all distinct values.
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One-Shot, Repeated Games and Communication
In this chapter, I will consider (without any great detail), how the
possibility of repeated interactions and communication may affect
behaviour.
In a one-shot game, players assume that they will not interact
again.
In a repeated game, players assume that they interact many times
and can adapt their behaviour to the previous behaviour of the
other player.
When players can communicate, they can co-ordinate their actions,
i.e. they are not forced to take their actions independently of each
other. Actions can be co-ordinated e.g. based on the result of a
coin toss.
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Pure Nash Equilibria in 2 × 2 Symmetric Games
Result 1: All 2 × 2 symmetric games have at least one pure Nash
equilibrium.
This can be proved by leading to a contradiction. Assume there
are no pure Nash equilibria.
Since (A, A) and (B, B) are not Nash equilibria, we have c > a and
b > d.
It follows from this that both (B, A) and (A, B) are Nash equilibria.
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Pure Nash Equilibria in 2 × 2 Symmetric Games
Result 2: There can be at most two pure equilibria, which appear
on a diagonal of the matrix describing the game.
Suppose (A, A) is a Nash equilibrium.
Hence, c < a. It follows that neither (A, B) nor (B, A) can be
Nash equilibria.
Now suppose (B, A) is a Nash equilibrium.
Hence, c > a and b > d. It follows that neither (A, A) nor (B, B)
can be Nash equilibria (the rest follows from an analogous
argument).
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
Assume that i) c > a, d > b, also ii) d > a, d > c.
The first pair of inequalities state that action B dominates action
A.
It follows that the strategy A can be eliminated and hence (B, B)
is the only Nash equilibrium.
From the final pair of inequalities, (B, B) Pareto dominates all of
the other payoff vectors, i.e. both players have a greater payoff
using this pair of strategies, than another of pair of strategies.
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
Game 1 in the set of experimental games was such a game
A
B
A (2, 2) (4, 6)
B (6, 4) (8, 8)
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
Switching the first three inequalities around, i.e.
c < a, d < b, d < a, and assuming a > b, we obtain a game in
which the (A, A) is the unique, Pareto dominant Nash equilibrium.
In such a case, the only rational action to take is the one
corresponding to the Nash equilibrium.
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
A variant of such games can be obtained by assuming b < d < c,
such that b+c
2 > d.
For example, the game
A
B
A (2, 2) (6, 14)
B (14, 6) (8, 8)
In this case, when no communication (co-ordination) is possible,
since B dominates A and (B, B) Pareto dominates (A, A), it seems
natural that in a one-shot game (B, B) will be played.
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
However, when the game is repeated, by co-ordinating their
actions, the players can obtain a higher mean reward by switching
between (B, A) and (A, B).
In this case, the players obtain an average reward of 10 per play
( 14+6
2 ), rather than 8.
In such cases, players use a so called ”tit-for-tat” (I scratch your
back, you scratch mine) strategy.
Players alternate between (A, B) and (B, A) until at least one of
them breaks such an agreement, from which point onwards the
equilibrium of the ”one-shot” game (B, B) is played.
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
It should be noted that in the one-shot game with communication,
the players can both obtain an expected reward of 10 by agreeing
to toss a coin and play (A, B) if heads appear, otherwise play
(B, A).
However, there is a serious problem with regards to such an
agreement.
If the toss suggests that (A, B) is played, then Player 1 has an
incentive to unilaterally defect from such an agreement (by playing
B he will obtain 8 rather than 6).
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a) A Single Pareto-Dominant Symmetric Pure Equilibrium
Similarly, if the toss suggests that (B, A) is played, then Player 2
has an incentive to unilaterally defect from such an agreement (by
playing B he will obtain 8 rather than 6).
Hence, reciprocation rather than communication is key to
obtaining the highest possible average payoff in such games.
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b) The Prisoner’s Dilemma
Assume that i) c > a, d > b, also ii) d < a.
The first pair of inequalities state that action B dominates action
A.
It follows that the strategy A can be eliminated and hence (B, B)
is the only Nash equilibrium.
However, (A, A) Pareto dominates (B, B). It follows that players
can arrive at socially non-optimal solutions by acting in their own
interests.
The Prisoner’s Dilemma is used to model situations in which there
are costs and benefits to cooperating. Each individual would prefer
the benefits of cooperation, without incurring the costs of
cooperation themselves.
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b) The Prisoner’s Dilemma
For example, consider the game (Game 3 on the experimental
games sheet)
A
B
A (5, 5) (0, 8)
B (8, 0) (2, 2)
In this case, the unique Nash equilibrium is (B, B), which gives
both players a payoff of 2. However, if both players chose A, then
they would both get a payoff of 5.
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b) The Prisoner’s Dilemma
In such a Prisoner’s dilemma, the action A is labelled
”cooperation” and the action B is labelled ”defection” (reversing
the inequalities above, the roles of the actions are swapped).
It is furthermore assumed that 2a > b + c, i.e. the players obtain
more by always cooperating than by alternating between (A, B)
and (B, A).
In such a case, ”tit-for-tat” strategies can lead to conditional
cooperation when interactions (games) are repeated.
As before, communication is not useful in a one-shot game to
achieve cooperation, since each player has an incentive to defect.
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c) Anti-Coordination Games
In this case, c > a and b > d. The Hawk-Dove game is an
anti-coordination game in which it is additionally assumed that two
”doves” (who play B) obtain a greater payoff than two ”hawks”
(who play A), i.e. d > a.
There are two pure Nash equilibria (A, B) and (B, A). Hence, the
name anti-coordination, since both players would prefer to choose
the action that the other player does not choose.
There is also a mixed Nash equilibria where both players use the
mixed strategy qA + (1 − q)B, where
q=
b−d
.
(c − a) + (b − d)
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c) Anti-Coordination Games
The Hawk-Dove game is used to model aggression.
Doves share resources.
Each player would prefer the other to be non-aggressive and then
by being aggressive, an aggressive player can gain resources.
However, the worst situation occurs when both players are
aggressive.
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c) Anti-Coordination Games
For example, the game (Game 4 on the experimental games sheet)
A
B
A (1, 1) (10, 4)
B (4, 10) (7, 7)
There are three Nash equilibria in this game (A, B), (B, A) and a
mixed equilibrium where A and B are chosen with equal
probabilities.
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c) Anti-Coordination Games
In one-shot games without communication, it is unclear how the
players should play unless some asymmetry is introduced.
In the natural world, such an asymmetry can based on
size/hierarchy or on the possession of a territory (owners are
aggressive, intruders are not).
When players are averse to risk, then it seems reasonable to play B
(dove) which ensures a payoff of at least 4, rather than A (hawk)
which can lead to a ”fight” (payoff of 1).
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c) Anti-Coordination Games
If communication is possible, the players could agree to play (A, B)
if a coin toss results in heads, otherwise play (B, A).
In this case, such an agreement is stable, since either player would
lose by unilaterally changing their action (both of these action
pairs are Nash equilibria).
It is best to avoid randomized strategies, due to the possible costs
of two-sided aggression.
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c) Anti-Coordination Games
In a repeated version of such a game, the players could alternate
between (A, B) and (B, A).
In addition, both players choosing B (dove) is stable, as long as
there is the threat of switching to A (hawk) in retaliation to the
other player choosing A.
Such strategies are called ”tit-for-tat” strategies, i.e. one player
continues to cooperate, as long as the other player cooperates.
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d) Coordination Games
In this case, a > c, d > b.
There are two Nash equilibria (A, A) and (B, B). Hence, the name
of such games, since both players prefer to choose the same action
as the other.
There is also a mixed equilibrium where both players use the mixed
strategy qA + (1 − q)B. where
q=
d −b
.
(a − c) + (d − b)
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d) Coordination Games
In such one-shot games, the behaviour that we expect from
rational players depends on the payoffs obtained at the two
equilibria and the risk associated with both actions.
(A, A) payoff dominates (B, B) if and only if a > d. It risk
dominates (B, B) if and only if a − c > d − b.
(B, B) payoff dominates (A, A) if and only if a < d. It risk
dominates (A, A) if and only if a − c < d − b.
The mixed equilibrium is always payoff dominated by both pure
equilibria.
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d) Coordination Games
The action A of a player minimax dominates an action B of the
same player, if the minimum payoff obtainable from taking action
A is greater than the minimum payoff obtainable from playing B.
If one pure equilibrium (A1 , A2 ) both risk dominates and payoff
dominates the other, where the action A1 minimax dominates the
other actions of Player 1 and the action A2 minimax dominates the
other actions of Player 2, then we expect players to choose the
dominant equilibrium.
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d) Coordination Games
For example, the game (Game 2 on the experimental games sheet)
A
B
A (10, 10) (4, 6)
B (6, 4) (7, 7)
There are three Nash equilibria in this game (A, A), (B, B) and a
mixed equilibrium where A and B are chosen with probabilities 3/7
and 4/7, respectively.
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d) Coordination Games
The Nash equilibrium (A, A) payoff dominates the equilibrium
(B, B) (a = 10 > d = 7).
In addition, the risk associated with A (3/7) is lower than the risk
associated with B (4/7) [a − c = 10 − 6 > d − b = 7 − 4].
However, for both players, action B (minimum possible payoff 6)
minimax dominates A (minimum possible payoff 4). Hence, it is
unclear what action rational players will take.
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d) Coordination Games
Another example is given by Game 5 on the experimental games
sheet)
A
B
A (5, 5) (−5, 4)
B (4, −5) (2, 2)
There are three Nash equilibria in this game (A, A), (B, B) and a
mixed equilibrium where A and B are chosen with probabilities 7/8
and 1/8, respectively.
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d) Coordination Games
The Nash equilibrium (A, A) payoff dominates the equilibrium
(B, B) (a = 5 > d = 2).
However, the risk associated with A (7/8) is much higher than the
risk associated with B (1/8) [a − c = 5 − 4 < d − b = 2 − (−5)].
Hence, it is unclear what strategy will be played by rational players.
The very large risk and the possible ”losses” associated with A
(minimum payoff -5) would suggest that many players would
choose B.
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d) Coordination Games
Intuitively, coordination games are easy to solve using
communication.
The players should simply agree to play the payoff maximising
equilibrium (neither has any incentive to change their action).
Similarly, in repeated games, it seems natural that the payoff
maximising equilibrium should be played, since by eventually
coordinating on this equilibrium will give a bettter sum of payoffs
than playing the other equilibrium (at least in the long run).
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e) A Two Player Game with no Pure Equilibria
As shown at the beginning of this chapter, every symmetric 2 × 2
game has at least one pure Nash equilibrium.
However, if a game is asymmetric, i.e. the players have
specific/different roles, then it is possible that no pure equilibrium
exists.
For example, consider the matching pennies game in which both
players choose action A or B.
Player 1 wins if he chooses the same action as Player 2, otherwise
Player 2 wins.
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e) A Two Player Game with no Pure Equilibria
In this case, the payoff matrix can be given by
A
B
A (1, 0) (0, 1)
B (0, 1) (1, 0)
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e) A Two Player Game with no Pure Equilibria
In this case, it is intuitively clear that neither of the players wishes
the other to know what action will be taken, hence no pure
equilibrium exists.
Note that if (A, A) or (B, B) is played, then Player 2 wishes to
switch.
Otherwise, if (A, B) or (B, A) is played then Player 1 wishes to
switch.
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e) A Two Player Game with no Pure Equilibria
Intuitively, at equilibrium, both players should choose each action
with probability 0.5.
This can be shown as follows:
Assume that Player 1 uses pA + (1 − p)B at equilibrium. In this
case, Player 2 is indifferent between actions A and B.
Playing A, Player 2 has an expected reward of
p × 0 + (1 − p) × 1 = 1 − p = v2 .
Playing B, Player 2 has an expected reward of
p × 1 + (1 − p) × 0 = p = v2 .
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e) A Two Player Game with no Pure Equilibria
At equilibrium v2 = p = 1 − p ⇒ p = 0.5, v1 = 0.5.
Analogously, assume that Player 2 uses qA + (1 − q)B at
equilibrium. In this case, Player 1 is indifferent between actions A
and B.
Playing A, Player 1 has an expected reward of
q × 1 + (1 − q) × 0 = q = v1 .
Playing B, Player 1 has an expected reward of
q × 0 + (1 − q) × 1 = 1 − q = v1 .
Thus, at equilibrium v1 = q = 1 − q ⇒ q = 0.5, v2 = 0.5.
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e) A Two Player Game with no Pure Equilibria
Hence, at equilibrium both players choose an action at random.
The unique value of the game is (0.5, 0.5).
It should be noted that a symmetric 3 × 3 game may have no pure
Nash equilibria.
For example, the Rock-Scissors-Paper game has no pure
equilibrium. Choosing an action at random is the only Nash
equilibrium in this game.
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