Exercise Exercise 1. Consider the Cournot game G = (I, (Ai, Ti)i≤1) such that: 1. I = {1,2} is the set of firms. 2. AR+ is the set of firm i's strategies. 3. TAR is firm i's payoff function defined by, for some c ≥ 0, Ti(Qi, 9−i) = max{1 − (qi + q−i), 0}qi — cqi. Answer the following questions: 1. Find all (pure-strategy) Nash equilibria in the case of c = = 0. 2. Find all (pure-strategy) Nash equilibria in the case of c = (0, 1). 3. Find all (pure-strategy) Nash equilibria in the case of c = [1,∞). Exercise 2. Consider the Braess's paradox G 1. I = {1,2,. , 10} is the set of people. ... 2. A = {1,0} is person i's strategy space. = (I, (Ai, ui)i≤1) such that: 3. u AR is person i's payoff function defined by -11-Σjel aj if ai =1 ui (a) = -11-Σje(1a) if a = 0, where Σ;a; is the number of people choosing strategy 1 and Σ; (1 - a;) is the number of people choosing strategy 0. Show that every (pure-strategy) Nash equilibrium has five people choosing route SAG and five players choosing route SBG.
Exercise Exercise 1. Consider the Cournot game G = (I, (Ai, Ti)i≤1) such that: 1. I = {1,2} is the set of firms. 2. AR+ is the set of firm i's strategies. 3. TAR is firm i's payoff function defined by, for some c ≥ 0, Ti(Qi, 9−i) = max{1 − (qi + q−i), 0}qi — cqi. Answer the following questions: 1. Find all (pure-strategy) Nash equilibria in the case of c = = 0. 2. Find all (pure-strategy) Nash equilibria in the case of c = (0, 1). 3. Find all (pure-strategy) Nash equilibria in the case of c = [1,∞). Exercise 2. Consider the Braess's paradox G 1. I = {1,2,. , 10} is the set of people. ... 2. A = {1,0} is person i's strategy space. = (I, (Ai, ui)i≤1) such that: 3. u AR is person i's payoff function defined by -11-Σjel aj if ai =1 ui (a) = -11-Σje(1a) if a = 0, where Σ;a; is the number of people choosing strategy 1 and Σ; (1 - a;) is the number of people choosing strategy 0. Show that every (pure-strategy) Nash equilibrium has five people choosing route SAG and five players choosing route SBG.
Chapter1: Financial Statements And Business Decisions
Section: Chapter Questions
Problem 1Q
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Transcribed Image Text:Exercise
Exercise 1. Consider the Cournot game G = (I, (Ai, Ti)i≤1) such that:
1. I = {1,2} is the set of firms.
2. AR+ is the set of firm i's strategies.
3. TAR is firm i's payoff function defined by, for some c ≥ 0,
Ti(Qi, 9−i) = max{1 − (qi + q−i), 0}qi — cqi.
Answer the following questions:
1. Find all (pure-strategy) Nash equilibria in the case of c = = 0.
2. Find all (pure-strategy) Nash equilibria in the case of c = (0, 1).
3. Find all (pure-strategy) Nash equilibria in the case of c = [1,∞).
Exercise 2. Consider the Braess's paradox G
1. I = {1,2,. , 10} is the set of people.
...
2. A = {1,0} is person i's strategy space.
=
(I, (Ai, ui)i≤1) such that:
3. u AR is person i's payoff function defined by
-11-Σjel aj
if ai
=1
ui (a)
=
-11-Σje(1a) if a = 0,
where Σ;a; is the number of people choosing strategy 1 and Σ; (1 - a;) is the number
of people choosing strategy 0.
Show that every (pure-strategy) Nash equilibrium has five people choosing route SAG and five
players choosing route SBG.
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