Question

Country A uses a fixed exchange rate system. There are 1 million financial traders, and each trader

chooses whether to sell the currency of country A simultaneously, i.e., each trader’s strategy is

either “sell” or “not.” If the number of traders who sell the currency is 0.5 million or larger, it is

known that the country devalues the currency. The followings are more details about the game.

• If a trader sells the currency, and the currency is devalued by the country, the trader gets

$1000.

• If a trader sells the currency, but the currency is not devalued, the trader loses $1 as the

trading cost.

• If a trader does not sell the currency, and the currency is not devalued, she does not gain or

lose any money.

• If a trader does not sell the currency, and the currency is devalued, she loses $100.

• Assume that each trader’s payoff is her wealth. We restrict our attention to pure strategy

Nash equilibria.

1. Is there a Nash equilibrium in which no one sells the currency? Answer by “yes” or “no.”

____________

2. Find a Nash equilibrium in which the currency is devalued (Just write the number of sellers

in the equilibrium).

The number of sellers in the equilibrium is ___________________________

3. What is the smallest number of sellers that is consistent with a Nash equilibrium?

The smallest number is ________________

4. Is there a Nash equilibrium in which equilibrium payoffs of some traders are different from

some other traders? Answer by “yes” or “no.” If your answer is “yes,” provide such a Nash

equilibrium.

_______________

Answer 1

1) Let us consider a scenario of no one selling currency. Now if a trader wants to deviate and sell his currency, then he will suffer a loss of $1 trading cost as only him selling currency doesn't devalue it. Hence there is no benefit for him to unilaterally change his strategy. Hence this scenario is a Nash equilibrium.

Answer : YES

2) Let us consider a scenario of everyone selling their currency. Since the number of traders selling is 1 million, the currency is devalued and everyone earns $1000. Now if a person would want to deviate his strategy to not sell, he will suffer a loss of $100. Hence he doesn't benefit from unilaterally deviating from the strategy. Hence this is a Nash equilibrium.

Answer: 1 million

3) As proven in question 1, the smallest number of sellers consistent with a Nash equilibrium is 0.

Answer: zero

4) No, there exists no Nash equilibrium where equilibrium payoffs of some traders are different from some other traders. Consider all cases.

Let us start with 0 sellers. All get a payoff of 0. This is a Nash equilibrium.

Let us increasing to 1 seller. He gets a payoff of -1 while others get a payoff of zero. This is not a Nash equilbrium as he can choose to not sell and get a payoff of 0.

Let us now have 2 sellers. Similar to the above case, both will benefit from unilaterally changing their strategy to not selling. Hence this is NOT a Nash equilibrium.

This will continue till we have 49,999,999 sellers.

Let us now have 49,999,999 sellers. All sellers earn a payoff of -1 and non sellers earn a payoff of 0. If a non seller unilaterally changes his strategy to selling, the number of sellers hits 50 million and the currency is devalued. Hence all the sellers earn a payoff of $1,000. Hence this is not a Nash equilibrium.

Let us now have 50 million sellers. Each seller earns a payoff of $1,000 and non sellers get -$100 payoff. Now every non seller is incentivised to unilaterally change his strategy to selling and earn a better payoff. Hence this is still not a Nash equilibrium.

This will continue till there is only one non seller left.

Let us have 1 non seller. He is benefitted from unilaterally changing his strategy to sell and this is still not a Nash equilibrium.

Let us consider the case where everyone sells. This is a Nash equilibrium and everyone earns a payoff of $1000.

Hence there exist only two Nash equilibria, one where nobody sells and one where everybody sells, and in both of them, every trader earns the same. Hence the answer is NO.

Answer: NO