ECON 323 Exam 3

In a two-player game, consider two strategy profiles (A, X) and (B, Y). According to the definition of Pareto domination, we say the profile of strategies (A, X) Pareto dominates (B, Y) if and only if _______.

(A, X) is at least as good as (B, Y) for both players and (A, X) is better than (B, Y) for at least one player By definition of Pareto domination, the profile of strategies (A, X) Pareto dominates (B, Y), if (A, X) is at least as good as (B, Y) for both players, and (A, X) is better than (B, Y) for at least one player.

In R-P-S, if played with sequential-moves where player 1 knows player 2's move, Is this strategy profile a subgame perfect Nash equilibrium?

Yes Each player plays the best strategy according to backward reduction.

We do not consider randomization. The relationship between a Nash equilibrium and an equilibrium in dominant strategies is that

an equilibrium in dominant strategies is a special case of a Nash equilibrium. An equilibrium in dominant strategies is always a Nash equilibrium, but the reverse is not true. Both Nash equilibrium and equilibrium in dominant strategies may not exist.

By definition, a Nash equilibrium occurs if and only if

each player is doing the best it can, given its opponents' strategy. In a Nash equilibrium, each player plays the best response given opponents' strategy.

In a Nash equilibrium

players may or may not have dominant strategies. In Nash equilibrium, players may or may not have dominant strategies. In prisoner's dilemma, both players have dominant strategy. But for example, in duopoly game shown in this homework, no player has dominant strategy.

In comparing the Cournot equilibrium with the competitive equilibrium

profit is higher, and output level is lower in Cournot Cournot equilibrium has lower output level and high price than competitive equilibrium.

Relative to the Cournot equilibrium, the Bertrand equilibrium with homogeneous products ____.

results in a larger output at a lower price Bertrand equilibrium has the same output and price as the competitive equilibrium. Hence, Bertrand equilibrium has a larger output and a lower price than Cournot equilibrium.

A dominant strategy produces ________.

the highest payoff for every possible strategy of the other players. A dominant strategy produces the highest payoff among all strategies of the agent, regardless of the strategy of his/her opponents.

The two features of a Prisoner's Dilemma

there is an equilibrium in dominant strategies the equilibrium is not Pareto efficient

Collusion on a high price can earn higher prices and higher profits under the Bertrand model, but why is this an unlikely outcome in practice?

Each collusive firm has an incentive to gain market share at the expense of the other firms by cutting prices. In the language of game theory, collusion Pareto dominates the Bertrand equilibrium for firms. But collusion is not a Nash equilibirum: each firm has the incentive to lower the price to increase market share.

The equilibrium in Nash equilibrium must be Pareto efficient.

False The Nash equilibrium may not be Pareto efficient. For example, the prisoner's dilemma.

In R-P-S, is (rock, paper) a Nash equilibrium?

No (Rock, Paper) is not a Nash equilibrium. Because given Player 2 plays paper, it is a best response for Player 1 to play scissors.

In R-P-S, is (paper, paper) a Nash equilibrium?

No Knowing that Player 2 plays "paper", Player 1's best response is to play "rock".

In comparing the Cournot equilibrium with the monopoly outcome (or perfectly collude case),

Total output level is higher in Cournot equilibrium; price level is higher in monopoly market. Monopoly market has a higher price and a lower total output level compared to Cournot equilibrium.

The Bertrand equilibrium ____.

has the same social welfare with competitive equilibrium Bertrand equilibrium has the same output, price, and welfare as the competitive equilibrium. As competitive equilibrium is efficient and has zero dead-weight loss, Bertrand equilibrium is efficient and has zero dead-weight loss.

In the sequential version of a game using the same players, the same strategies, and the same possible outcomes as the original game, the subgame perfect Nash equilibrium ________.

may be different than the Nash equilibrium in the original game. They may be different. This can be seen from the duopoly game shown in this homework. They may be same. This can be seen from prison's dilemma for example.