Game Theory with Applications to Finance and Marketing, I Solutions to Homework 3 You can choose to work on any two problems below and submit your solution sheets to TA on November 12. 1. (Currency Attack.) This exercise concerns the strategic interactions between the government of a nation and a group of speculators in the foreign exchange market.1 The state of the economy is denoted by θ, which is uniformly distributed over the unit interval [0; 1]. The exchange rate in the absence of government intervention is f(θ), with f 0(·) > 0, so that a higher realization of θ represents a stronger state of the economy. The exchange rate is initially pegged by the government at e∗. Facing the government is a continuum of identical speculators located at the unit interval: for each a 2 [0; 1], there is exactly one speculator located at a. Thus the total population of speculators is one. Each speculator may either attack the currency by selling short one unit of the currency, or refrain from trading. Short-selling costs t to the speculator. If a speculator short-sells the currency and the government abandons the exchange rate peg, then the speculator's payoff is e∗ − 1The following story can be found on the internet: The Quantum Group of Funds are privately owned hedge funds based in Curacao (Netherlands Antilles) and Cayman Islands. They are currently advised by George Soros through his company Soros Fund Management. Soros started the fund in 1973 in partner- ship with Jim Rogers. The shareholders of the funds are not publicly disclosed although it is known that the Rothschild family and other wealthy Europeans put $6 million into the funds in 1969. In 1992, the lead fund, Soros' Quantum Fund, became famous for 'breaking' the Bank of England, forcing it to devalue the pound. Soros had bet his entire fund in a short sale on the ultimately fulfilled prediction that the British currency would drop in value, a coup that netted him a profit of $1 billion. In 1997, Soros was blamed for forcing sharp devaluations in Southeast Asian currencies. In July 2011, the Quantum Fund announced they would be ending the fund, and will be returning all outside money by the end of 2011. The fund will now exclusively manage Soros' family money. 1 f(θ) − t; but if the government defends the peg, then the speculator gets −t. Assume that2 e∗ ≥ f(1) > e∗ − t ≡ f(θ) > f(0): If the speculator chooses not to attack the currency, his payoff is zero. The government's payoff from abandoning the exchange rate is zero, while its payoff from defending the exchange rate is v − c(α; θ), where α is the population of speculators who choose to attack the currency, v is a positive constant representing the value of maintaining the peg, and c(·; ·) is continuously differentiable and such that • @c @c 3 @α > 0 > @θ ; • c(1; 1) > v;4 • c(0; θ) < c(0; θ) ≡ v < c(0; 0).5 The game proceeds as follows. • All speculators and the government first observe the realization of θ. • Then, all speculators simultaneously decide whether to spend c and attack the currency. • Then the government learns the population α of speculators that have chosen to attack the currency. The government then decides to or not to abandon the peg. The game ends after the government makes the decision. 2That is, given that the government would defend the peg, attacking the currency is worthwhile to a speculator if and only if the state of economy is weaker than or equal to θ. 3That is, it is less costly for the government to defend the currency if the economy is in a stronger state or if there are fewer speculators choosing to attack the currency. 4That is, when all speculators choose to attack the currency, the government's cost of defending the currency exceeds its value, regardless of the state of the economy. 5That is, when the economy is in the weakest state, it is never worthwhile for the government to defend the peg; and when no speculators choose to attack the currency, there exists a benchmark state θ < θ of the economy such that the government would defend the peg if and only if θ ≥ θ. 2 Assume that a speculator will not attack the currency if he feels in- different about attacking or not attacking, and that the government will abandon the peg if it feels indifferent about abandoning or not abandoning the peg. (i) Suppose that θ 2 [0; θ]. Does the government abandon the peg in equilibrium? Determine the equilibrium α. (ii) Suppose that θ 2 (θ; θ). Does the government abandon the peg in equilibrium? Determine the equilibrium α.6 (iii) Suppose that θ 2 [θ; 1]. Does the government abandon the peg in equilibrium? Determine the equilibrium α. 2 @c @c Solution. Consider part (i). For θ [0; θ], we have by @α > 0 > @θ that v = c(0; θ) ≤ c(0; θ) ≤ c(α; θ); 8α 2 [0; 1]; and hence it is the government's dominant strategy to not defend the peg. Rationally expecting that the government will not defend the peg, a speculator can get e∗ − t − f(θ) = f(θ) − f(θ) > 0 by attacking the currency, since f 0(·) > 0 and θ ≤ θ < θ: Thus in equilibrium we have α = 1, and the government will abandon the peg with probability one. Next, consider part (iii). Given that θ 2 [θ; 1], a speculator would get 6Unlike in part (i) and part (iii), there may be multiple equilibria in part (ii). 3 e∗ − t − f(θ) = f(θ) − f(θ) ≤ 0 by attacking the currency, and hence no speculator chooses to attack the currency. Thus we have α = 0 in equilibrium. It follows from @c @c @α > 0 > @θ and θ ≥ θ > θ that v = c(0; θ) > c(0; θ) ≥ c(0; θ); and hence the government chooses to defend the peg in equilibrium. Finally, consider part (ii). We claim that the equilibria obtained in respectively part (i) and part (iii) are both equilibria in part (ii). Indeed, if a single speculator expects all his fellow speculators will at- tack the currency, so that α = 1, then since v < c(1; 1) < c(1; θ); 8θ 2 (θ; θ); the government will abandon the peg upon seeing α = 1. It follows that by joining his fellow speculators to attack the currency, that single speculator would get e∗ − t − f(θ) = f(θ) − f(θ) > 0; which implies that the single speculator had better follow suit. This proves that there does exist an equilibrium with α = 1. On the other hand, if a single speculator expects all his fellow specula- tors will not attack the currency, so that α = 0,7 then since 7With a continuum of speculators (represented by the unit interval), each single spec- ulator is of zero Lebesgue measure. In other words, the equilibrium α is independent of the equilibrium behavior of a single speculator. 4 v = c(0; θ) > c(0; θ); 8θ 2 (θ; θ); the government will defend the peg upon seeing α = 0, which implies that the single speculator had better follow suit and not attack the currency either. This proves that there does exist an equilibrium with α = 0. If we remove the assumption that the government will abandon the peg if it feels indifferent about abandoning or not abandoning the peg, and if we assume instead that the government may adopt a mixed strategy in the latter situation, then there also exists an equilibrium for part (ii) where the government randomizes between defending and abandoning the peg. Given θ 2 (θ; θ), we can define two constants a(θ) and π(θ) as follows. By the monotonicity and continuity of c(·; θ), and by the intermediate value theorem, there must exist a unique a(θ) 2 (0; 1) such that c(1; θ) > v = c(a(θ); θ) > c(0; θ); since v < c(1; 1) < c(1; θ); 8θ 2 (θ; θ); and v = c(0; θ) > c(0; θ); 8θ 2 (θ; θ): Now, define π(θ) 2 (0; 1) as the solution to the following equation (1 − π)[f(θ) − f(θ)] − πt = 0 f(θ) − f(θ) ) π(θ) = : f(θ) − f(θ) + t 5 Then, for each θ 2 (θ; θ), there exists an equilibrium where (A) all speculators feel indifferent about attacking or not attacking the cur- rency, and in equilibrium a group of speculators with population a(θ) choose to attack the currency; and (B) upon seeing α = a(θ), the gov- ernment feels indifferent about defending or abandoning the peg, and in equilibrium the government may defend the peg with probability π(θ). Remarks. This exercise is taken from Morris and Shin (1998, Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks, American Economic Review). The multiplicity of equilibrium in part (ii) is dis- turbing, because the government cannot develop effective anti-attack strategies without knowing which equilibrium will prevail. The au- thors show that if the speculators do not observe θ directly but instead each receives a private signal about θ, then the problem of equilibrium multiplicity disappears. For this reason, the government has reasons to withhold some detailed information about the true state of the economy from the public investors. We shalll come back to this point after we talk about games with incomplete information. Observe that the like- lihood of currency attacks decreases when the economy grows stronger; in particular, as shown in part (iii), when θ ≥ θ, we have α = 0 in equi- librium.