課程名稱︰商業賽局模型一 課程性質︰選修 課程教師︰陳其美 開課學院：管理學院 開課系所︰財金所 考試日期（年月日）︰2010/12/15 考試時限（分鐘）：18:30-21:30 (180m) 是否需發放獎勵金：是 （如未明確表示，則不予發放） 1.Consider the following two-player game: Player 1/Player 2│ L│ M│ R ─────────┼───┼───┼──── U│ 40,0 │ 0,20│-10,-800 ─────────┼───┼───┼──── D│ 20,10│ 10,0 │ 0,30 Let π be the probability that player 1 choose U in equilibrium, and p and q respectively the probabilities that player 2 choose L and M in equilibrium. (i)Find (π,p,q) for the unique pure-strategy Nash equilibrium. (ii)Show that this game has two mixed-strategy Nash equilibria. Find (π,p,q) for each mixed-strategy Nash equilibrium. 2.Consider the following extensive game, called G(1). In this game G(1), there are two dates (date 1 and date 2), and two players (a bank B and a borrowing firm F). At date 1, F has no money, it has two mutually exclusive projects at hand, and it needs to borrow 2 dollars from B to invest in either of them. Project S is riskless and it can generate Y dollars at the end of date 1. Project R is risky, and F must incur a personal disutility -k<0 at the beginning of date 1 if F decides to invest on project R. If project R is taken, then at the end of date 1, with probability π it may generate X dollars and with probability 1-π it may generate nothing, but when the latter unpleasant outcome occurs, if B is willing to lend another 2 dollars to F at the beginning of date 2, the the project R can be continued for one more period, and it will generate r dollars for sure at the end of date 2, together with a non-monetary private benefit u>0 to F.1 At each date t=1,2, B has exactly 2 dollars for lending. Moreover, we assume that at the beginning of date 1, B cannot commit to lending F any money at date 2; B will lend F money at date 2 only if doing so is optimal (or subgame perfect) for B at date 2. To simplify tha analysis, assume that the following numerical values are valid: X=12=u, Y=4, π=1/2 r=1 k=5 Finally, assume that F will choose project S whenever F feels indirrerent about project S and project R. The timing of G(1) can be summarized as follows. *At date 1, after F asks B to lend it 2 dollars, B must choose the face value of the debt D≧0.2 *The game ends if F rejects B's offer D. If F accepts B's pffer, then F must choose between project R and project S. Choosing project R incur a disutility -k to F at this point. *Then at the end of date 1, the date-1 cash flow z is realized, where z=Y if F has chosen project S in the previous stage, and z may equal either X or zero if F has chosen project R in the previous stage. Given z, B gets min(D,z), and F gets z-min(D,z)=max(z-D,0). *Then at the beginning of date 2, B can decide whether to lend another 2 dollars to F and change the face value of debt from D to D'≧0.3 The game ends if B choose not to lend 2 dollars the second time. *If B has chosen to lend another 2 dollars to F at the previous stage, then at the end of date 2, the date-2 cash flow is realized, and B gets min(D',r). At this point, F gets r-min(D',r)=max(r-D',0) together with the private benefit u. (i)Assume that over the two-date period, F and B are risk-neutral about monetary payoffs and they have no time preferences (so that the payoffs obtained at the two dates can be added together without discounting). Find the SPNE of the above extensive game. Which project is chosen in equilibrium? What is the equilibrium D? What is the bank's equilibrium payoff? What is F's equilibrium payoff?4 (ii)Now, consider the infinitely repeated version G(∞) of the above stage game G(1), where at stage n=1,2,..., F and B must play G(1) repeatedly. Assume that F and B have a common discount factor 0<ρ<1 that applies to two consecutive stages, although we still assume that within each stage (or within each G(1)), there is no discounting for F and B. Find the smallest ρ* such that if ρ is greater than ρ* then there is an SPNE sustained by the trigger strategy, where F invests in project R at the first date in each stage n, and B will lend another 2 dollars at the second date of each stage n if and only if project R generate no cash at the first date in stage n.5 3.Reconsider the game chain-store paradox with two entrant E1 and E2. We shall modify the crazy incumbent's payoffs as follows. Suppose that entry of a new firm may result in a positive network effect if the incumbent is of the crazy type. For this reason we assume that the crazy incumbent get 3/2 when it accomodates a new entrant, and it gets 2 when it preys a new entrant. As in lecture 4, the (crazy or sane) incumbent get 3/4 in a period without entry, and the sane incumbent gets 0 and -1 when it, respectively, accomodates and preys a new entrant; and an entrant gets zero by staying out, and if entry oucurs then the entrant get 1 and -1 respectively after the incumbent chooses to accomodate and prey. Find all the PBEs of this modified chain-store paradox with two entrants.6 Footnotes: 1.Only monetary payoffs such as X,Y and r can be shared with B. The private benefit u cannot be given to B although it is also measured in monetary terms. 2.If F accepts this offer, then F gets 2 dollars from B at the beginning of date 1, and F must repay B the minimum of D and whatever it has at the end of date 1---we assuming that F is protected by limited liability. 3.This is obviously not gonna happen if F has chosen project S, or if F has chosen project R, and project R has successfully generated cash flow X. 4.Hint:First consider the subgame where at the beginning of date 2, B is considering whether it should lend another 2 dollars to F, after B has lent F the first 2 dollars, and after F has chosen project R which generated the date-1 earnings z=0. Should B lend another 2 dollars to F in this subgame? Now, move backward to consider F's investment decision at date 1, given that F has accepted a loan contract specifying a face value D. Determine F's optimal investment decision for non-negative D. Now, move backward again to consider B's choice of D at the beginning of date 1. What is B's optimal choice of D? Which project will F choose to implement given this optimal D? What is B's equilibrium payoff? What is F's equilibrium payoff? 5.Hint:For part (ii), conjecture that in the best SPNE from the bank's perspective, at each stage the bank will lend F another 2 dollars in case F chooses project R in that stage and produces a date-1 cash flow z=0. The bank may be tempted to deviate at date 2 in each stage n. Verify that the immediate gain from such a deviation is 1, and the loss in each future stage is the difference between the bank's profit obtained in part (i) and 7/2. Solve for ρ* by letting the bank's no-deviation IC constraint binding at ρ*. 6.Hint:The sane type has a dominant strategy in this game. Solve this game by backward induction. First consider the subgame where E1 has just entered the industry. Can this subgame have a separating PBE? Can it have a pooling PBE? Can it have a semi-pooling PBE? Then move backward to consider E1's entry decision at the earlier stage. Show that the equilibrium path depends on whether 3/4＜x1≦1, or 1/2＜x1≦3/4, or 0≦x1≦1/2. --

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