NPTEL MOOC Game Theory 2016: Solution #1
The game table for Second-Price Auction is given below.
0
10
20
30
40
50
0
40,0
0,30
0,30
0,30
0,30
0,30
10
40,0
30,0
0,20
0,20
0,20
0,20
20
40,0
30,0
20,0
0,10
0,10
0,10
30
40,0
30,0
20,0
10,0
0,0
0,0
40
40,0
30,0
20,0
10,0
0,0
0,-10
50
40,0
30,0
20,0
10,0
0,0
-10,0
The valuations of the players are v1 = 40, v2 = 30 for players 1, 2 respectively. Consider now the
outcome (10,10) i.e. both players bid 10. Since there is a tie, player 1 wins the auction. He pays
the second highest bid, which is also 10. Therefore, his net payoff is v1-10 = 40 -10 =30. Payoff
to player 2 is 0 since he loses the auction. Rest of the entries can be explained similarly. The
answers to the various questions are given below.
1. u1(10,20) is payoff to player 1 when the bids are 10, 20 for players 1, 2 respectively. As
can be seen from the table above, this is 0.
Ans d
2. u2(40,20) is payoff to player 2 when the bids are 40, 20 for players 2, 1 respectively. As
can be seen from table above, this is 10.
Ans b
3. Best responses of player 2 to bid 20 of player 1 can be seen to be 30, 40 and 50 since all
of them yield the same payoff i.e. 10. Out of the answers given, only 30 is correct. Hence,
Ans c
4. Best responses of player 1 to bid 20 of player 2 are 20, 30, 40, 50. Hence all of the above
Ans d
5. The bid 0 by player 1, is weakly dominated by bids 10, 20, 30. For example, if player 2
bids 10, bid of 0 by player 1 yields 0, while bid of 10 yields 30. Similarly, you can see
that for all other bids of player 2, bid of 10 either yields equal or better payoff. Same is
true for bids 20, 30 of player 1.
Ans d
6. Consider outcome (10,20). Payoff to player 1 is 0. He has an incentive to unilaterally
deviate to 20 and increase his payoff to 20. Therefore (10,20) is NOT a Nash equilibrium.
Similarly in (20,20) player 2 has an incentive to shift to 30. In (40,50) player 2 has an
incentive to shift to 40. Therefore, only Nash equilibrium is (40,30) from which it can be
seen that no one has an incentive to deviate unilaterally
Ans a
7. (40,50) yields payoffs 0,-10 to players 1, 2 respectively. Both can improve payoff by
shifting to (10, 10) for instance, hence it is NOT Pareto optimal. Similarly (50,50) is
NOT Pareto optimal since it yields -10,0 and both can improve payoffs by shifting to
(20,40) for instance.
Ans c.
The game table for the Market Game is shown below.
0
1
2
3
4
5
0
0.0
0,4
0,6
0,6
0,4
0,0
1
4,0
3,3
2,4
1,3
0,0
0,0
2
6,0
4,2
2,2
0,0
0,0
0,0
3
6,0
3,1
0,0
0,0
0,0
0,0
4
4,0
0,0
0,0
0,0
0,0
0,0
5
0,0
0,0
0,0
0,0
0,0
0,0
The price per unit is p = max{5-(q1+q2), 0}. Consider the outcome (1,2) where q1 and q2 are 1,2
respectively. The price per unit is max{2,0} = 2. Therefore, payoff to player 1 is 1  2 = 2 and
payoff to player 2 is 2  2 = 4. When q1 = 3 and q2 = 4, price per unit p = max{-2,0}. Hence
payoff to each player is 0. Rest of the entries can be derived similarly.
8. u1(3,1) is payoff to player 1 for quantities 3,1 of firms 1, 2 respectively. From table
above, this can be seen to be 3.
Ans a
9. u2(3,0) can be seen to be payoff to player 2 for quantities 3,0 of players 2, 1 respectively.
From table above this can be seen to be 6.
Ans c
10. The best response quantity of Firm 1 to quantity q2 = 1 of Firm 2 can be seen be to q1 = 2,
since this yields the highest payoff of 4 to player 1, higher than the rest of the actions.
Ans c
11. It can be seen that all response of Firm 2 yield the same payoff i.e. 0 to quantity q1 = 4 of
Firm 1. Hence any action is a best response.
Ans c.
12. It can be seen that no Firm has an incentive to deviate unilaterally from either (2,2), (2,1)
or (1,2). Hence all above are Nash equilibria.
Ans d
13. It can be seen that (4,4), (4,5) and (5,4) are all Nash equilibria since each yields payoff of
0 to both players and remains 0 on any unilateral deviation.
Ans d
14. It can be seen that Nash equilibrium (2,1) is Pareto optimal since there is no other
outcome which yields higher payoff to both firms. On the other hand, the Nash
equilibrium (5,5) is NOT Pareto optimal since it yields 0 to both players and both can
increase payoff by shifting for instance to (1,1). Therefore, at least one of the Nash
equilibria is Pareto optimal i.e. (2,1)
Ans c
15. As described in solution 14 above, Nash equilibrium (2,1) is Pareto optimal. Further, the
outcome (1,1) is NOT a Nash equilibrium. For instance Firm 1 can deviate to quantity q1
= 2 to increase payoff from 3 to 4. However, it can be seen to be Pareto optimal since
there is no other outcome which yields a higher payoff to both players. Hence, the only
option true is that at least one of the Pareto optimal points is a Nash equilibrium.
Ans c