Mixed Strategy
Equilibrium
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vNM Preferences in Strategic Games: Complete, Transitive, Continuous, Symmetric, Lecture notes of Game Theory
The assumptions made about preferences in Chapters 2 and 3, including completeness, transitivity, and continuity. The document also introduces strategic games with vNM preferences, which consist of a set of players, actions for each player, and preferences regarding lotteries over action profiles. the definition of a mixed strategy Nash equilibrium and its properties, as well as symmetric games and strictly dominated strategies.
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Mixed Strategy
Equilibrium
Matching Pennies
Christos A. Ioannou
Heads Tails
Heads
Tails
Player 1
Player 2
- (^) The ordinal preferences for player 1 are (H, H) ∼ 1 (T, T ) 1 (H, T ) ∼ 1 (T, H).
- (^) SGWOP has no (pure) Nash Equilibrium.
- (^) A lottery is a probability distribution over outcomes.
- (^) To talk about steady state in this game, we need to define preferences over lotteries.
Previous Assumptions on Preferences
- (^) In Chapters 2 and 3, we assumed preferences are:
- (^) Complete; that is, we can compare all lotteries, and
Christos A. Ioannou
Previous Assumptions on Preferences
- (^) In Chapters 2 and 3, we assumed preferences are:
- (^) Complete; that is, we can compare all lotteries, and
- (^) Transitive; that is, if A B and B C, then, A C.
Christos A. Ioannou
Previous Assumptions on Preferences
- (^) In Chapters 2 and 3, we assumed preferences are:
- (^) Complete; that is, we can compare all lotteries, and
- (^) Transitive; that is, if A B and B C, then, A C.
- (^) With these assumptions, we can guarantee that there is a
payoff function such that,
A B ⇐⇒ u(A) > u(B).
- (^) Suppose that an individual has preferences A B C.
Christos A. Ioannou
Previous Assumptions on Preferences
- (^) In Chapters 2 and 3, we assumed preferences are:
- (^) Complete; that is, we can compare all lotteries, and
- (^) Transitive; that is, if A B and B C, then, A C.
- (^) With these assumptions, we can guarantee that there is a
payoff function such that,
A B ⇐⇒ u(A) > u(B).
- (^) Suppose that an individual has preferences A B C.
- (^) Suppose an action can guarantee B.
Christos A. Ioannou
Previous Assumptions on Preferences
- (^) In Chapters 2 and 3, we assumed preferences are:
- (^) Complete; that is, we can compare all lotteries, and
- (^) Transitive; that is, if A B and B C, then, A C.
- (^) With these assumptions, we can guarantee that there is a
payoff function such that,
A B ⇐⇒ u(A) > u(B).
- (^) Suppose that an individual has preferences A B C.
- (^) Suppose an action can guarantee B.
- (^) What if I can randomize between A with probability 0. and C with probability 0.5.
- (^) To compare these, we need a Christos A. Ioannou
Previous Assumptions on Preferences
- (^) In Chapters 2 and 3, we assumed preferences are:
- (^) Complete; that is, we can compare all lotteries, and
- (^) Transitive; that is, if A B and B C, then, A C.
- (^) With these assumptions, we can guarantee that there is a
payoff function such that,
A B ⇐⇒ u(A) > u(B).
- (^) Suppose that an individual has preferences A B C.
- (^) Suppose an action can guarantee B.
- (^) What if I can randomize between A with probability 0. and C with probability 0.5.
- (^) To compare these, we need a cardinal utility function. Christos A. Ioannou
More Assumptions on Preferences
- (^) Now we make two more assumptions about lotteries over
outcomes. Specifically, that preferences exhibit:
- (^) Continuity; that is, if A B C, then, there exists some p ∈ [0, 1] such that,
Christos A. Ioannou
More Assumptions on Preferences
- (^) Now we make two more assumptions about lotteries over
outcomes. Specifically, that preferences exhibit:
- (^) Continuity; that is, if A B C, then, there exists some p ∈ [0, 1] such that, pA + (1 − p) C = B, and
Christos A. Ioannou
More Assumptions on Preferences
- (^) Now we make two more assumptions about lotteries over
outcomes. Specifically, that preferences exhibit:
- (^) Continuity; that is, if A B C, then, there exists some p ∈ [0, 1] such that, pA + (1 − p) C = B, and
- (^) Independence; that is, if A B, then, for any N and any p ∈ [0, 1], pA + (1 − p) N pB + (1 − p) N.
Christos A. Ioannou
More Assumptions on Preferences
- (^) Now we make two more assumptions about lotteries over
outcomes. Specifically, that preferences exhibit:
- (^) Continuity; that is, if A B C, then, there exists some p ∈ [0, 1] such that, pA + (1 − p) C = B, and
- (^) Independence; that is, if A B, then, for any N and any p ∈ [0, 1], pA + (1 − p) N pB + (1 − p) N.
- (^) With these, we can guarantee that there is a payoff
function such that, for any two lotteries
L = pA + (1 − p) B and M = pC + (1 − p) D,
Christos A. Ioannou
von Neumann-Morgenstern
Preferences
- (^) If preferences satisfy Completeness, Transitivity,
Continuity, and Independence, then, the preferences can
be represented with a cardinal utility function.
Christos A. Ioannou
von Neumann-Morgenstern
Preferences
- (^) If preferences satisfy Completeness, Transitivity,
Continuity, and Independence, then, the preferences can
be represented with a cardinal utility function.
- (^) Preferences that satisfy the four conditions are called von
Neumann and Morgenstern (vNM) preferences.
Christos A. Ioannou