A large number n of investors make a joint investment (of 1 unit each) which yields...

 

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A large number n of investors make a joint investment (of 1 unit each) which yields a return r > 1 (per unit invested) if it matures. For the project to mature requires that at least L, 1 < L < n, investors do not withdraw their money early. Yet, each investor may experience an interim liquidity shock which forces her to withdraw. More formally, an investor may be one of two types. Type 1 of investor i occurs with a small probability 0, making i's payoff function u₂1 (s) = 1 - sį, where si € {0, 1}. With probability 1-0, investor i is of type 2 and has payoff function U₁2 (s): = TS₁ if Σ=18;2L 1-si if Σ-18; <L Each investor learns her own type, but not the type of other investors, and decides whether to withdraw (s₁ = 0) or not (s; = 1) conditional on her type. Assume that L-1 T 1 + > Σ (1) 0² (1-0) (0-4) 0)(n-k) 1+r k=1 (a) Write down a "rough" extensive form of this game, assuming n = 2 and L = 1. (b) Find a (Bayesian) Nash equilibrium of this game. (For the general case.) A large number n of investors make a joint investment (of 1 unit each) which yields a return r > 1 (per unit invested) if it matures. For the project to mature requires that at least L, 1 < L < n, investors do not withdraw their money early. Yet, each investor may experience an interim liquidity shock which forces her to withdraw. More formally, an investor may be one of two types. Type 1 of investor i occurs with a small probability 0, making i's payoff function u₂1 (s) = 1 - sį, where si € {0, 1}. With probability 1-0, investor i is of type 2 and has payoff function U₁2 (s): = TS₁ if Σ=18;2L 1-si if Σ-18; <L Each investor learns her own type, but not the type of other investors, and decides whether to withdraw (s₁ = 0) or not (s; = 1) conditional on her type. Assume that L-1 T 1 + > Σ (1) 0² (1-0) (0-4) 0)(n-k) 1+r k=1 (a) Write down a "rough" extensive form of this game, assuming n = 2 and L = 1. (b) Find a (Bayesian) Nash equilibrium of this game. (For the general case.)

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Here is the rough extensive form of the game assuming n 2 and L 1 1 Player 1 Type 1 Withdraw s1 0 Stay s1 1 2 Player 2 Type 1 Withdraw s2 0 Stay s2 1 ... View the full answer