Game Theory with Applications to Finance and Marketing

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At first, note that exerting a high effort is Mr. A’s equilibrium strategy if it is common knowledge that Mr. A will not sell shares by the end of date 2, because 1 > β > 1 and 0 < c < 2. 4 This can never be Mr. A’s equilibrium strategy if it is common knowledge that Mr. A will sell all his shares at date 1. To see this, suppose instead that it were. Then by exerting a high effort, Mr. A expects that π 1 = 0 (and hence he can get the proceeds π 1 + π 2 = 5 by selling the firm) with probability β and π 1 = 1 (and hence he can get the proceeds π 1 + π 2 = 1 by selling the firm) with probability 1 − β, so that Mr. A’s payoff is accordingly 5β + (1 − β) − c = 1 + 4β − c. However, by exerting a low effort, Mr. A can make sure that π 1 = 0, so that he can sell the firm for a price of 5 for sure at date 1. Thus Mr. A will always deviate, which is a contradiction to the supposed equilibrium. Having shown that exerting a high effort cannot be Mr. A’s equilibrium behavior, I must emphasize that this does not imply that exerting a low effort is automatically Mr. A’s equilibrium behavior. Suppose that Mr. A does exert a low effort in equilibrium. Then he gets nothing by selling his firm. However, by exerting a high effort, Mr. A expects that π 1 = 1 with probability 1 − β, and following such a zero-probability event the public investors have no choice but to believe that the firm is worth at least one dollar (because π 2 is always nonnegative, regardless of Mr. A’s choice of effort). 19 Hence if c < 1 − β, exerting a low effort is not Mr. A’s equilibrium behavior either. We conclude that there are no conditions on c and β that can support an equilibrium where Mr. A exerts a high effort, and that c > 1 − β 19 The market does not know that Mr. A has deviated, and hence they still price the firm at zero upon seeing zero profit in period 1. Only when the period-1 profit is 1, the market is forced to re-assess the situation, and the market can then infer that this can only happen after Mr. A has exerted a high effort, and hence the market also infers that the period-2 profit is zero. Thus by exerting a high effort, the manager spends c, but only gets 1 with prob. 1 − β. For Mr. A’s exerting a low effort to be part an NE, we conclude that c must be greater than 1 − β. 34
is the necessary and sufficient condition that supports the equilibrium where Mr. A exerts a low effort. This exercise delivers the basic idea of signal-jamming. Note that if the market believes that Mr. A will exert a high effort (because it is efficient to do so from the firm’s perspective), then Mr. A has an incentive to fool the market and raise his own payoff by exerting a low effort. By definition, a Nash equilibrium is a state of decision-making by all players, where nobody predicts his rivals’ actions incorrectly. Hence in equilibrium, Mr. A cannot have a chance to fool the market, and we have shown that this implies a corporate inefficiency in the current exercise—only exerting a low effort is consistent with a Nash equilibrium of the current game. 29. Example 10. Consider a firm run by an owner-manager Mr. A. The firm has 1 share of common stock outstanding. At date 1, Mr. A can either costlessly exert a low effort (e = 0) or exert a high effort (e = 1) by incurring a (non-monetary) disutility c > 0. Mr. A’s effort choice is unobservable to public investors. Let π 1 and π 2 be respectively the firm’s date-1 and date-2 profits. If Mr. A exerts a high effort, the firm’s date-1 and date-2 profits are both 2. If Mr. A exerts a low effort, then the firm’s date-1-date-2 profits may be (π 1 , π 2 ) = { (0, 1), with probability β; (2, 6), with probability 1 − β. The firm has no growth opportunities after date 2, and it will be liquidated at the end of date 2. Mr. A and all investors in the financial market are risk-neutral without time preferences. Now suppose that the firm has decided not to distribute cash dividends at date 1, and that β = 5 7 , c = 2. (i) Suppose that it is common knowledge that Mr. A will never sell shares by the end of date 2. What is the date-1 share price of the firm? (ii) Suppose that, contrary to part (i), it is common knowledge that Mr. A will sell all the equity at date 1, and that public investors’ only date-1 information about the firm is π 1 . What is the date-1 share price 35
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Game Theory with Applications to Finance and Marketing

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