Term
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Definition
(Strategic game with ordinal preferences)
A strategic game (with ordinal preferences) consists of:
- for each player, a set of actions
- for each player, preferences over the set of action profiles.
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Term
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Definition
(Nash equilibrium of strategic game with ordinal preferences)
The action profile a* in a strategic game with ordinal preferences is a Nash equilibrium if, for every player i and every action ai of player i, a* is at least as good according to player i's preferences as the action profile (ai , a*-i) in which player i chooses ai while every other player j chooses a*j . Equivalently, for every player i,
ui(a*) ≥ ui(ai , a*-i) for every action ai of player i,
where ui is a payoff function that represents player i's preferences. |
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Term
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Definition
In a strategic game with ordinal preferences,
player i's action a" i strictly dominates her action a'i if
ui(a" i , a-i) > ui(a'i , a-i) for every list a-i of the other players' actions,
where ui is a payoff function that represents player i's preferences. We say that the
action a0i is strictly dominated. |
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Term
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Definition
In a strategic game with ordinal preferences,
player i's action a" i weakly dominates her action a'i if
ui(a" i , a-i) ≥ ui(a'i, a-i) for every list a-i of the other players' actions
and
ui(a" i , a-i) > ui(a'i , a-i) for some list a-i of the other players' actions,
where ui is a payoff function that represents player i's preferences. We say that the
action a'i is weakly dominated. |
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Term
| Symmetric two-player strategic game with ordinal preferences |
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Definition
A two-player strategic game with ordinal preferences is symmetric if the players' sets of actions are the same and the players' preferences are represented by payoff functions u1 and u2 for which u1(a1, a2) = u2(a2, a1) for every action pair (a1, a2). |
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Term
| Symmetric Nash equilibrium |
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Definition
An action profile a* in a strategic game with ordinal preferences in which each player has the same set of actions is a symmetric Nash equilibrium if it is a Nash equilibrium and a*i is the same for every player i.
(Note: A symmetric game may have no symmetric Nash equilibrium)
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Term
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Definition
| A mixed strategy of a player in a strategic game is a probability distribution over the player's actions. |
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Term
Mixed strategy Nash Equilibrium
of strategic game with vNM preferences) |
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Definition
The mixed strategy profile a* in a stratic game with vNM preferences is a (mixed strategy) Nash equlibrium if, for each player i and every mixed strategy ai of player i, the expected payoff to player i of a* is at least as large as the expected payoff to player i of (ai, a*-1) according to a payoff function whose expected value represents player i's preferences over lotteries. Equivalently, for each player i,
Ui(a*)≥Ui(ai,a*-i) for every mixed strategy ai of playeri,
where Ui(a) is player i's expected payoff to the mixed strategy profile.
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Term
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Definition
For a strategic game with ordinal preferences, Bi(a-i) is the set of player i's best actions when the list of the other players' actions is a-i. For a strategic game with vNM preferences, Bi(a-i) is the set of player i's best mixed strategies when the list of the other players' mixed strategies is a-i.
(Note: the mixed strategy profile a* is a mixed strategy NE iff a*i is in Bi(a*-i) for every player i. |
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| Characterization of mixed strategy NE of finite game |
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Definition
A mixed strategy profile a* in a strategic game with vNM preferences in which each player has finitely many actions is a mixed strategy NE iff, for each player i,
- the expected payoff, given a*-i, to every action to which a*i assigns positive probability is the same
- the expected payoff, given a*-i, to every action to which a*i assigns zero probability is at most the expected payoff to any action to which a*i assigns positive probability.
Each player's expected payoff in an equilibrium is her expected payoff to any of her actions that she uses with positive probability. |
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Term
Existence of mixed strategy Nash equlibrium in finite games
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Definition
| Every strategic game with vNM preferences in which each player has finitely many actions has a mixed strategy NE. |
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Term
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Definition
In a strategic game with vNM preferences, player i's mixed strategy ai strictly dominates her action a'i, if
Ui(ai,a-i) > ui(a'i,a-i) for every list a-i of the other players' actions,
where ui is a payoff function whose expected value represents player i's preferences over lotteries and Ui(ai,a-i) is player i's expected payoff under ui when she uses the mixed strategy ai and the actions chosen by the other players are given by a-i. We say that the action a'i is strictly dominated.
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Term
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Definition
In a strategic game with vNM preferences, player i's mixed strategy ai weakly domintes her action a'i if
Ui(ai,a-i) ≥ ui(a'i,a-i) for every list a-i of the other players' actions
and
Ui(ai,a-i) > ui(a'i,a-i) for some list a-i of the other players' actions,
where ui is a payoff function whose expected value represents player i's preferences over lotteries and Ui(ai,a-i) is player i's expected payoff under ui when she uses the mixed strategy ai and the actions chosen by the other players are given by a-i. We say that the action is weakly dominated.
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Term
| Existence of mixed strategy NE with no weakly dominated strategies in finite games (Proposition 122.1) |
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Definition
| Every strategic game with vNM preferences in which each player has finitely many actions has a mixed strategy NE in which no player's strategy is weakly dominated. |
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Term
| Pure strategy equilibria survive when randomnization is allowed (Proposition 122.2) |
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Definition
| Let a* be a NE of G and for each player i let a*i be the mixed strategy of player i that assigns probability one to the action a*i. Then a* is a mixed strategy NE of G'. |
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Term
| Pure strategy equilibria survive when randomnization is prohibited (Proposition 123.1) |
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Definition
| Let a* be a mixed strategy NE of G' in which the mixed strategy of each player i assigns probability one to the single action a*i. Then a* is a NE of G. |
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Term
| Symmetric two-player strategic game with vNM preferences (Definition 129.1) |
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Definition
| A two-player strategic game with vNM preferences is symmetric if the players' sets of actions are the same and the players' preferences are represented by the expected values of payoff functions u1 and u2 for which u1(a1,a2) = u2(a2,a1) for every action pair (a1,a2). |
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Term
| Symmetric mixed strategy Nash equilibrium (Definition 129.2) |
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Definition
| A profile a* of mixed strategies in a strategic game with vNM preferences in which each player has the same set of actions is a symmetric mixed strategy NE if it is a mixed strategy NE and a*i is the same for every player i. |
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| Existence of symmectric mixed strategy NE in symmetric finite games (Proposition 130.1) |
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Definition
| Every symmetric strategic game with vNM preferences in which each player's set of actions is finite has a symmetric mixed strategy NE. |
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Term
| Characterization of mixed strategy NE (Proposition 142.2) |
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Definition
A mixed strategy profile a* in a strategic game with vNM preferences is a mixed strategy NE iff, for each player i,
- a*i assigns probability zero to the set of actions ai for which the action profile (ai,a*-i) yields player i an expected payoff less than her expected payoff to a*
- for no action ai does the action profile (ai, a*-i) yield player i an expected payoff greater than her expected payoff to a*.
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