Microeconomics I: Game Theory Lecture 7: Dominance Concepts (see Osborne, 2009, Sect 2.7.8,2.9,4.4) Dr. Michael Trost Department of Applied Microeconomics December 6, 2013 Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 1 / 44
Strict Nash equilibrium In this lecture, we introduce alternatives to the Nash equilibrium concept. The first one is a refinement of the Nash equilibrium concept and is called strict Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 2 / 44
Dominance concepts After that, we present two solution concepts which are referred to as strict dominance and weak dominance. Both concepts are discussed in context with (ordinary) strategic games and with strategic games with von Neumann- Morgenstern preferences. Moreover, relationships between these dominance concepts and the (mixed action) Nash equilibrium concept are explored. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 3 / 44
Let s play! But before we dip into theory, let us play a game which is known as the PUBLIC GOODS GAME. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 4 / 44
Example: Public Goods Game Each of the n subjects (n > 2) receive an amount of e 10. Every subject secretly chooses how many of this amount to put into a public pot. It is required that the player s contribution to the pot is a nonnegative integer number. The total amount in this pot is doubled and then evenly divided among all subjects. The final income of each subject is the amount of money the subject kept plus the amount of money obtained from the public pot. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 5 / 44
Example: Public Goods Game EXERCISE : Specify the components of the PUBLIC GOODS GAME. ANSWER: Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 6 / 44
Example: Public Goods Game EXERCISE : Find the Nash equilibrium of the PUBLIC GOODS GAME by the two-step method! ANSWER: 1 Best response correspondence of player i {1,..., n} B i (a i ) = 2 Nash equilibrium of the PUBLIC GOODS GAME IS a := (a i ) i {1,...,n} = Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 7 / 44
Example: Public Goods Game The Nash equilibrium of the PUBLIC GOODS GAME has the property that every player is worse off if she deviates from her Nash equilibrium action given that the other players stick to their Nash equilibrium actions. In general, a Nash equilibrium having such property is called strict. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 8 / 44
Strict Nash equilibrium Definition 7.1 (Strict Nash equilibrium) Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action profile a := (ai ) i I A is a strict Nash equilibrium of Γ if, for every player i I, U i (ai, a i) > U i (a i, a i) holds for every action a i A i \ {ai }. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 9 / 44
Strict Nash equilibrium The strict Nash equilibrium is a Nash equilibrium in which every player is worse off if she deviates from her equilibrium action given that the other players follow their equilibrium actions. It is a refinement of the (ordinary) Nash equilibrium concept. Indeed, the Nash equilibrium concept only demands that every player is not better off (but maybe equally well) if she deviates from her equilibrium action but if the other players stick to their equilibrium actions. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 10 / 44
Exercise: Secrecy Game Player B Player A don t talk talk don t talk 1,1 0,0 talk 0,0 0,0 EXERCISE: Figure out all Nash equilibria of the SECRECY GAME. Which of them are strict? Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 11 / 44
Exercise: Public Goods Game EXERCISE: Show that the Nash equilibrium of the PUBLIC GOODS GAME is strict! EXPLANATION: Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 12 / 44
Dominance concepts In the following, we introduce the solution concepts of strict dominance and weak dominance. These concepts are defined in context with (ordinary) strategic games and in context with strategic games with von Neumann-Morgenstern preferences. As we already know, in the latter form of a strategic game the preferences of the players are extended to the set of all lotteries over the action profiles. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 13 / 44
Strict dominance Definition 7.2 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i a i A i if A i of player i I strictly dominates her action U i (a i, a i ) > U i (a i, a i ) holds for every list a i A i of the other players action. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 14 / 44
Strictly dominant action Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I is termed strictly dominant if it strictly dominates every other action a i A i \ {a i } of player i. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 15 / 44
Exercise: Prisoner s Dilemma Suspect B Suspect A quiet fink quiet 2,2 0,3 fink 3,0 1,1 EXERCISE: Do the suspects have strictly dominant actions in the PRISONER S DILEMMA? Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 16 / 44
Exercise: Public Goods Game EXERCISE: Do the subjects of the PUBLIC GOODS GAME have a strictly dominant action? If yes, which one? EXPLANATION: Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 17 / 44
Strict dominance and Nash equilibrium Theorem 7.3 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). If action a j A j of some player j I is strictly dominant, then every Nash equilibrium a := (a i ) i I of Γ has the property a j = a j. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 18 / 44
Strict dominance and strict Nash equilibrium Corollary 7.4 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ) and suppose each player i I has a strictly dominant action a i A i. Then action profile a := (a i ) i I is the unique Nash equilibrium of Γ and it is even a strict Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 19 / 44
Application of Theorem 7.4 Consider the PUBLIC GOODS GAME. It has been shown that the action a i :=... of every player i {1,..., n} is strictly dominant. By Corollary 7.4, we conclude that action profile a := (a i ) i {1,...,n} is the unique Nash equilibrium of the PUBLIC GOODS GAME and, moreover, it is even a strict Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 20 / 44
Strictly dominated action Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I is termed strictly dominated if there is some other action a i dominates her action a i. A i \ {a i } of player i that strictly Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 21 / 44
Exercise: Game Γ 1 Strategic Game Γ 1 Player B Player A left middle right up 1,3 1,0 2,1 middle 1,1 3,1 1,2 down 0,2 0,2 1,3 EXERCISE: Do the players of game Γ 1 have actions that are strictly dominated. If yes, which ones? Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 22 / 44
Strictly dominated action Theorem 7.5 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). If a := (a i ) i I is a Nash equilibrium of Γ, then action a i of every player i I is not strictly dominated. This theorem says that a strictly dominated action is not used in any Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 23 / 44
Exercise: Game Γ 1 Strategic game Γ 1 Player B Player A left middle right up 1,3 1,0 2,1 middle 1,1 3,1 1,2 down 0,2 0,2 1,3 EXERCISE: Find the Nash equilibria of game Γ 1! Compare your result with that of the previous exercise. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 24 / 44
Weak dominance Definition 7.6 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I weakly dominates her action a i A i if the two conditions 1 U i (a i, a i) U i (a i, a i) holds for every list a i A i of the other players actions 2 U i (a i, a i) > U i (a i, a i) holds for some list a i A i of the other players actions are satisfied. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 25 / 44
Weakly dominant action Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I is termed weakly dominant if it weakly dominates every other action a i A i \ {a i } of player i I. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 26 / 44
Exercise: Secrecy Game Player B Player A don t talk talk don t talk 1,1 0,0 talk 0,0 0,0 EXERCISE: Do the players of the SECRECY GAME have a weakly dominant strategy? If yes, which one? Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 27 / 44
Weak dominance and Nash equilibrium Theorem 7.7 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). If each action of action profile a := (a i ) i I is weakly dominant, then action profile a is a Nash equilibrium of Γ. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 28 / 44
Weak dominance and Nash equilibrium Consider a strategic game in which each player has a weakly dominant action. Then, by Theorem 7.7, the action profile consisting of these actions constitutes a Nash equilibrium. However, it has not be the only one as be demonstrated by the SECRECY GAME. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 29 / 44
Example: Secrecy Game Player B Player A don t talk talk don t talk 1,1 0,0 talk 0,0 0,0 The SECRECY GAME has two Nash equilibria, namely (don t talk, don t talk) and (talk, talk). However, for both players, talking is not a weakly dominant action. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 30 / 44
Weakly dominated action Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I is termed weakly dominated if there is some other action a i dominates action a i. A i \ {a i } of player i that weakly Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 31 / 44
Weakly dominated action Theorem 7.8 Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). If action profile a := (a i ) i I of Γ is a strict Nash equilibrium, then every action a i of this profile is not weakly dominated. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 32 / 44
Example: Secrecy Game Consider the SECRECY GAME. Player B Player A don t talk talk don t talk 1,1 0,0 talk 0,0 0,0 As we already know, the action profile (don t talk, don t talk) is a strict Nash equilibrium of the SECRECY GAME. Theorem 7.8 implies that the actions of this Nash equilibrium are not weakly dominated. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 33 / 44
Dominance in vnm strategic games Now, we extend the ideas of strict and weak dominance to cases in which the players have von Neumann- Morgenstern preferences (so called vnm strategic games). In such games the players preferences are defined on the lotteries over the possible action profiles. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 34 / 44
Mixed action profiles containing pure actions Consider vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ) and let a i A i be a list of definite actions of players different to player i I. The mixed action profile (α i, a i ) j I (A j ) means that player i chooses the mixed action α i, every player j different to i chooses the definite a j. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 35 / 44
Mixed action profiles containing pure actions The expected utility U i of player i obtained by profile (α i, a i ) is ( ) U i (α i, a i ) = a A α j (a j) u i (a i, a i) j I = a i A i α i (a i) u i (a i, a i ) where the second row follows from the fact that, for each player j I \ {i}, mixed action α j is the degenerated lottery that ascribes probability of 1 to action a j. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 36 / 44
Strict dominance in vnm strategic games Definition 7.9 Consider a vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ). Player s i mixed action α i (A i ) strictly dominates her definite action a i A i if U i (α i, a i ) > u i (a i, a i ) holds for every list a i A i of the other players action. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 37 / 44
Strictly dominated in vnm strategic game Consider a vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I is termed strictly dominated in vnm strategic game Γ if there is some mixed action α i (A i ) of player i that strictly dominates action a i in Γ. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 38 / 44
Exercise: VNM strategic game Γ 2 VNM strategic game Γ 2 Player B Player A left right up 4,0 1,1 middle 2,2 2,2 down 1,1 4,0 EXERCISE: Is there an action that is strictly dominated in vnm strategic game Γ 2? If yes, which one? Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 39 / 44
Strictly dominated in vnm strategic games Theorem 7.10 Consider the vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ). If α := (α i ) i I is a mixed Nash equilibrium of Γ, then every action a i A i with α i (a i) > 0 is not strictly dominated in Γ. This theorem says that a strictly dominated action of a vnm strategic game is not used with positive probability in any mixed Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 40 / 44
Exercise: VNM strategic game Γ 2 VNM strategic game Γ 2 Player B Player A left right up 4,0 1,1 middle 2,2 2,2 down 1,1 4,0 EXERCISE: Does vnm strategic game Γ 2 has a Nash equilibrium in pure actions? Determine all its Nash equilibria in mixed actions. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 41 / 44
Weak dominance in vnm strategic games Definition 7.11 Consider a vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ). Player i s mixed action α i (A i ) is said to weakly dominate action a i A i if the two conditions 1 U i (α i, a i ) u i (a i, a i) holds for every list a i A i of the other players actions, 2 U i (α i, a i ) > u i (a i, a i) holds for some list a i A i of the other players actions are satisfied. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 42 / 44
Weakly dominated in vnm strategic games Consider a vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i A i of player i I is termed weakly dominated in vnm strategic game Γ if there is some mixed action α i (A i ) of player i that weakly dominates a i in Γ. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 43 / 44
Weakly dominated in vnm strategic games Theorem 7.12 Every finite vnm strategic game Γ := (I, (A i ) i I, ( i ) i I ) has a mixed action Nash equilibrium in which no weakly dominated action is used with positive probability. Dr. Michael Trost Microeconomics I: Game Theory Lecture 7 44 / 44