The Unofficial Solution Manual A Primer In Game Theory by Robert Gibbons
Unofficial Solutions Manual To R.a Gibbon's A Primer In Game Theory
Static Games of Complete Information
1.1 What is a game in normal form? What is a strictly dominated
strategy in a normal-form game? What is a pure-strategy Nash
equilibrium in a normal-form game?
Answer 1.1 See text.
1.2 In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibra?
Answer 1.2 B is strictly dominated by T. C is now strictly dominated by R. The strategies (T,M) and (L,R) survive the iterated elimination of strictly dominated strategies. The Nash Equilibrium are (T,R) and (M,L).
1.3 Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have s1 and s2, where 0 < si, s2 < 1. If Si + s2 < 1, then the players receive the shares they named; if si + s2 > 1, then both players receive zero. What are the pure-strategy Nash equilibria of this game?
Answer 1.3 For whatever value Individual 1 chooses (denoted by S1), Individual 2’s best response is S2 = B2(S1) = 1 - S2. Conversely, S1 = B1(S2) = 1 - S1. We know this because if S2 < 1 - S1, then there is money left on the table and Individual 2 could increase his or her payoff by asking for more. If, however, S2 > 1 -S1, Individual 2 earns nothing and can increase his payoff by reducing his demands sufficiently. Thus the Nash Equilibrium is S1 + S2 = 1.
Answer 1.4 The market price of the commodity is determined by the formula P = a ô€€€ Q in which Q is determined Q = q1 + ... + qn The cost for an individual company is given by Ci = c.qi. The profit made by a single firm is
1.2 In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibra?
Answer 1.2 B is strictly dominated by T. C is now strictly dominated by R. The strategies (T,M) and (L,R) survive the iterated elimination of strictly dominated strategies. The Nash Equilibrium are (T,R) and (M,L).
1.3 Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have s1 and s2, where 0 < si, s2 < 1. If Si + s2 < 1, then the players receive the shares they named; if si + s2 > 1, then both players receive zero. What are the pure-strategy Nash equilibria of this game?
Answer 1.3 For whatever value Individual 1 chooses (denoted by S1), Individual 2’s best response is S2 = B2(S1) = 1 - S2. Conversely, S1 = B1(S2) = 1 - S1. We know this because if S2 < 1 - S1, then there is money left on the table and Individual 2 could increase his or her payoff by asking for more. If, however, S2 > 1 -S1, Individual 2 earns nothing and can increase his payoff by reducing his demands sufficiently. Thus the Nash Equilibrium is S1 + S2 = 1.
Answer 1.4 The market price of the commodity is determined by the formula P = a ô€€€ Q in which Q is determined Q = q1 + ... + qn The cost for an individual company is given by Ci = c.qi. The profit made by a single firm is
A similar argument applies to all other firms.
Answer 1.5 Let qm be the amount produced by a monopolist. Thus, if the two were colluding, they’d each produce
Their profits are reversed when their production is. Thus, the payoffs
are:
been had they cooperated by producing qm together (which would have earned them 0.13 (a-c)2).
Answer 1.6 Price is determined by P = a-Q and Q is determined by Q = q1 + q2. Thus the profit generated for Firm 1 is given by:
Since quantities cannot be be negative. Thus, under certain conditions, a sufficient difference in costs can drive one of the firms to shut down.
Answer 1.7 We know that,
Therefore, this is not a Nash Equilibrium
If pi = pj = c, then pi = pj = 0. Neither firm has any reason to deviate; if FFirm i were to reduce pi, pi would become negative. If Firm i were to raise pi, qi = pi = 0 and he would be no better off. Thus Firm i (and, symmetrically, Firm j) have no incentive to deviate, making this a Nash Equilibrium.
Answer 1.8 The share of votes received by a candidate is given by
Answer 1.9 See text.
Answer 1.10 (a) Prisoner’s Dilemma
In a mixed strategy equilibrium, Player 1 would choose q such that Player 2 would be indifferent between Mum and Fink. The payoff from playing Mum and Fink must be equal. i.e.
-1.q + -9 . (1 - q) = 0 . q + -6 . (1 - q) ⇒ q = -3.5
This is impossible.
Here, Player 1 must set p so that Player 2 is indifferent between Left, Middle and Right. The payoffs from Left and Middle, for example, have to be equal. i.e.
In a mixed equilibrium, Player 1 sets p0 and p1 so that Player 2 would be indifferent between L, C and R. The payoffs to L and C must, for example, be equal i.e.
Answer 1.11 This game can be written as
Answer 1.12
Player 1 will set p such that
Answer 1.5 Let qm be the amount produced by a monopolist. Thus, if the two were colluding, they’d each produce
are:
Answer 1.6 Price is determined by P = a-Q and Q is determined by Q = q1 + q2. Thus the profit generated for Firm 1 is given by:
Answer 1.7 We know that,
If pi = pj = c, then pi = pj = 0. Neither firm has any reason to deviate; if FFirm i were to reduce pi, pi would become negative. If Firm i were to raise pi, qi = pi = 0 and he would be no better off. Thus Firm i (and, symmetrically, Firm j) have no incentive to deviate, making this a Nash Equilibrium.
Answer 1.8 The share of votes received by a candidate is given by
Answer 1.9 See text.
Answer 1.10 (a) Prisoner’s Dilemma
-1.q + -9 . (1 - q) = 0 . q + -6 . (1 - q) ⇒ q = -3.5
This is impossible.
Similarly, the payoffs from Middle and Right have to be equal
Which, besides contradicting the previous result, is quite impossible.
Answer 1.11 This game can be written as
In a mixed Nash Equilibrium, Player 1 sets p0 and p1 so that the expected payoffs from L and C are the same. i.e.
Player 2 will set q such that
Answer 1.13
There are two pure strategy Nash Equilibrium (Apply to Firm 1, Apply to Firm 2) and (Apply to Firm 2, Apply to Firm 1). In a mixed-strategy equilibrium, Player 1 sets p such that Player 2 is indifferent between Applying to Firm 1 and Applying to Firm 2.
Answer 1.14
Dynamic Games of Complete Information
Answer 2.1 The total family income is given by
Where B is the maximizing level of the bequest. We know it exists because a) there are no restrictions on B and b) V() and U() are concave and increasing
The child’s utility function is given by U(Ic(A) - B*(A)). This is maximized at
The child’s utility function is given by U(Ic(A) - B*(A)). This is maximized at
Answer 2.2 The utility function of the parent is given by V(Ip-B) + k[U1(Ic-S) + U2(B + S)]. This is maximized at
Answer 2.3 To be done
Answer 2.4
Answer 2.5
Given this condition, the worker will acquire the promotion iff he acquires the skill. He will acquire the skill iff the benefit outweighs the cost i.e.
Answer 2.6 The price of the good is determined by
P(Q) = a - q1 - q2 - q3
The profit earned by a firm is given by pi = (p - c) .qi. For Firm 2, for example
pi2 = (a - q*1- q2 - q3 - c) . q2
which is maximized at
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