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VARIATIONS ON SIMPLE PAYOFF MATRICES

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Title: VARIATIONS ON SIMPLE PAYOFF MATRICES

1
VARIATIONS ON SIMPLE PAYOFF MATRICES

2
The Payoff Matrix

Given any payoff matrix, the standard assumption
is
that the players choose their strategies
simultaneously, or
in any event, that each player chooses a strategy
in ignorance of the strategic choice of the other
player,
or, as described in the original Playing Games
handouts, by secret ballot, and
without pre-play communication.
However, it is enlightening to consider
variations on this standard setup, such as the
following.

3
The Payoff Matrix (cont.)

The answer to the question of whether a given
type of variation
makes a difference and
makes what kind of difference
itself varies greatly according to the nature of
the payoff matrix and the nature of the game it
represents.
That is, whether the game is Pure Coordination,
Battle of Sexes, Battle of Bismarck Sea , D-Day,
Prisoners Dilemma, Chicken, etc.
For simplicity, we continue to suppose that the
payoff matrix is 2 2,
i.e., there are two just two players,
each with a choice between two strategies,
so the game has just four possible outcomes
(cells in the matrix).
We suppose also that the payoff matrix is common
knowledge to each player, i.e.,
each player knows what the other players payoffs
(interests/preferences/values/goals are).

4
Sequential Choice with Perfect Information

Suppose that the players make their strategic
choices sequentially and openly,
producing a game with perfect information.
Note that this entails two variants of a given (2
2) matrix
player 1 makes the first move, and
player 2 makes the first move.
Does the fact that moves are made sequentially
affect the choice that either player makes?
Does it affect the outcome of the game?
If so, does the advantage go to the first-mover
or the second-mover?
Might both players benefit, or be hurt, as a
result of sequential (vs. simultaneous) moves?

5
Full Pre-play Communication

Suppose that the players can engage in
unrestricted pre-play communication before
choosing their strategies.
Does this affect their strategic choices?
Does either player have an incentive to
communicate his intentions truthfully?
Would a message necessarily be believed by the
other player?

6
Limited Pre-play Communication

Suppose that the players can engage in only
limited pre-play communication,
Specifically, that one player can send a single
one-way message to the other player (who cannot
reply) before they choose their strategies.
Would the privileged player send such a message?
Would it be truthful?
Would it be believed by the other player?

7
Strategic Intelligence

Suppose that one player gains strategic
intelligence, i.e.,
somehow finds out in advance the strategy
chosen by the other player in advance.
Is such strategic intelligence always useful to
the player who gains it?
Is it ever harmful to the player who gains it?
What might the other player do if he discovers
that his strategic plan have been found out?
Might the other player want to have his strategic
plan found out?

8
Strategic Deception

Suppose that one player may (attempt to) engage
in strategic deception, i.e.,
may allow the other player to apparently find
out his strategy in advance but this information
may be misleading.
Is it always advantageous to deceive the other
player in this way?
Can it ever be harmful?
What might the other player do if he discovers
that you are attempting to deceive him?

9
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10
Credible Unconditional Commitment

Suppose that a player can use the opportunity
for pre-play communication to (somehow) convey
(perhaps by means of some overt strategic move) a
credible or irrevocable unconditional commitment
to a strategy choice.
Might such a player commit himself to a different
strategy than he would otherwise choose?
Will this advantage the player who makes the
commitment?
Might it advantage the other player also?

11
Threats and Promises

Suppose that the players make sequential choices
but that the second-mover can use the opportunity
for pre-play communication to (somehow) convey an
credible or irrevocable conditional commitment to
a strategy choice, i.e.,
if you (the first-mover) choose your strategy X,
I will choose my strategy Y.
Might the second mover conditionally commit
himself to a different strategy than what he
would otherwise choose?
Will this conditional commitment take the form of
a threat or a promise?
Will this advantage the player who makes the
conditional commitment?
Might it advantage the other player also?

12
Cooperative Games and Side Payments

Suppose the players can use the opportunity for
pre-play communication
to negotiate and enter into a binding (or
enforceable) agreement as to what strategy each
will chose and
perhaps to reallocate their joint payoffs in some
agreed upon manner, i.e., to make side payments?
Does this affect their strategy choices and the
outcome of the games?

13
Repeated Play

Suppose that the game is iterated that is,
the same players will play this game repeatedly.
and
know that they will do so.
Does this affect their strategy choice and the
outcome of each play of the game?
Does matter whether the number of iterations is
known to the players?
Can the players acquire reputations with respect
to how they play?
Will these reputations help them in games with
other players.

14
An Unproblematic Zero-Conflict Game