VARIATIONS ON SIMPLE PAYOFF
MATRICES
Topic #6
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.
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, Prisoner’s 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 player’s payoffs
(interests/preferences/values/goals are).
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 secondmover?
• Might both players benefit, or be hurt, as a result of
sequential (vs. simultaneous) moves?
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?
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?
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”?
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?
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?
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?
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?
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.
An Unproblematic Zero-Conflict Game
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects
A Zero-Conflict Coordination Game
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects
A Coordination Game with Conflict of Interest
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects
A Strictly-Determined Zero-Sum Game
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects
A Non-Strictly Determined Zero-Sum Game
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects
A Prisoner’s Dilemma Game
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects
A Chicken Game
•
•
•
•
•
•
•
•
•
•
Sequential Choice
Communication
Intelligence
Deception
Commitment
Threats
Promises
Binding Agreements
Side Payments
Reputation Effects