The entire problem of why strong firms downsize, and why the equity of downsizing firms is bought was presented in Chapter IV as an information problem, and not one of traditional finance. In this section, I will relax the assumption of the information barrier, and will solve the game, assuming that Players 1 and 2 both know the state of the firm's health. In this game, Nature's move is observable not only to Player 1 (the firm) but also to Player 2 (the investor). All of the payoffs remain the same. This can be considered a situation in which the shareholders are as equally informed as the management about the company's economic prospects for the short run. While such an example is rare in the real world, scenarios can be imagined in which investors have very accurate perceptions about the state of a firm's health. In the game presented in Chapter IV, downsizing was not a signal for health since it did not alter the Bayesian probabilities of health. In this case, downsizing is also not a signal for health, since the investors know with perfect certainty what the health of the firm really is. This is indicated by the lack of multiple-node information sets for Player 2.

To solve this game, backward induction will be used. Player 2 must create a strategy for each node at which he must make a decision, even nodes that Player 1's strategy dictate will never be reached. This is to ensure that Player 2's strategy is credible, and that Player 1 will have no profitable deviations from his announced strategy, and vice versa. The analysis will begin with the decisions of Player 2.

Beginning with the top-left node, if Player 2 observes a strong firm downsizing, he will receive 0 if he sells, and 1 if he buys, so he will buy. If he observes a weak firm downsizing, he will sell the stock, since the benefit of owning the stock of a company with management that makes the right decision (1) is not as high as the benefit from selling the stock of a weak company (2). If he observes a strong firm not downsizing, he will certainly buy. There is no reason for the firm to downsize, so player 2 will have the best return possible: that of a strong company with management that makes good decisions. If Player 2 observes weak firms not downsizing, he will surely sell, since there is no benefit at all to owning a weak company that has management refusing to lay off workers. Player 2's decisions look as follows.

Given Player 2's strategy after every possible history of interaction between the two players, Player 1 must formulate a strategy. If the firm is weak, it knows that the investor will always sell, so it is better off by downsizing. Here, management receives no points for being sold, but earns one point for making the right decision if it downsizes. If the firm is strong, it knows that the investor will always buy, so it does not downsize. Not only does the firm receive two points for being strong, but also it obtains an extra point for making the right decision (not to downsize). The game now has a unique strategy profile, corresponding to one and only one Nash Equilibrium.

This analysis proves that neither player has an incentive to mix strategies, and is therefore a perfect separating equilibrium. Different types of Player 1 will always behave differently, as weak firms have no incentive to masquerade as strong firms, since they cannot hide their weakness from the market. The strategy profile looks as follows:

This game offers an interesting comparison of who wins and loses when the flow of information changes. Player 2 will receive a utility of 2-the maximum possible utility-no matter what happens. By observing not only Player 1's behavior, but also the choices of Nature, Player 2 is able to buy and sell strong and weak companies respectively. A strong Player 1 will also receive the maximum possible payoff. His stock will be bought, and he will make the right labor decision. Before the advent of perfect information, there were inefficiencies in which a strong Player 1 would not receive 3 points, since the market might perceive him as weak, and sell his stock, so the firm would be forced to downsize to maximize its expected benefits. Perfect information is clearly advantageous to strong firms, who have nothing to hide. Weak firms will always receive a payoff of 1 in this game. In the game in which investors were uncertain, weak players always benefited from the lack of information flow. When everyone was maintaining the work force, investors bought the stock, and the weak firm received 2 for convincing the market to buy its stock. It received 3 when everyone downsized, and not only was it able to convince the market to buy its stock, but also it got an extra point for making the right decision with respect to the labor force. In contrast, the best it could ever do in a game of perfect information is 1.

The equity value of a firm is clearly not only a function of whether or not a firm is strong or weak, and whether or not it downsizes, but is also a function of how good the information flow is between investors and management. Define an "imperfect market" as one in which one party cannot observe the state of nature. In this model, strong firms operating in an imperfect market are relatively undervalued to strong firms operating in a perfect market. In contrast, weak firms operating in an imperfect market are relatively overvalued to weak firms operating in a perfect market. The payoff to investors in never higher in an imperfect market compared to a perfect market.

The fact that the equity value of a firm is a function of the flow of information indicates that there exists a principal-agent dilemma in terms of asymmetric information between the management and the investors, as both of these players have different preferences with respect to labor choices and the health of the firm. The decrease (increase) in firm value for a strong (weak) firm in an imperfect market as opposed to a perfect market can be thought of as the agency cost (benefit) of the lack of information flow.

Now that the equity values of the firm in the case of perfect and imperfect information have been contrasted, the equity value of the firm under the assumption of no information will be calculated. In this case, not even the firm's management knows whether or not it is healthy. The firm must make a labor decision without knowing whether or not it will be weak or strong in the coming months. To facilitate comparison with the game developed in AT&T chapter, the probabilities of the firm being weak or powerful will be set at 20% and 80% respectively.

The probability that player 2's left information set will be reached is the probability that Player 1 will downsize. This is given as

This calculation is trivial, since Player 1 will downsize with the same probability, regardless of his strength. Likewise, the probability that Player 2's right information set will be reached is the probability that he will not downsize, which is given as . Assume player 2's left information set has been reached. He will then calculate the Bayesian probabilities for each node at his information set.

If , then p will be 0.8. Otherwise, it will be undefined. The payoff to selling will be

The payoff to buying will be

The investor will always buy whenever . If , then the probabilities do not exist, as the information set will never be reached in equilibrium. The investor is free to make up any probabilities that justify his actions since this information set will never be reached. Noting that the investor is indifferent when the chances of him being at either node are the same, he will arbitrarily choose any value of p between 0 and 1 inclusive, and this will determine his strategy.

Similar arguments can be used to forecast the investor's actions if his right information set is reached. Player 2 will calculate the Bayesian probabilities of being at each node in his information set.

If , then q will be 0.8. Otherwise, it will be undefined. The payoff to selling will be

The payoff to buying will be

The investor will always buy whenever . If , then the probabilities do not exist, as the information set will never be reached in equilibrium. The investor is free to make up any probabilities that justify his actions. Noting that the investor is indifferent when the chances of him being at either node are the same, he will arbitrarily choose any value of q between 0 and 1 inclusive, and this will determine his strategy.

Likewise, Player 1 will be making decisions based on the probabilities of Player 2 buying or selling the stock, as well as the probability that he is strong or weak. The payoff to downsizing is given as the following: the expected payoff to downsizing if Player 1 is weak times the probability that he is weak plus the expected payoff to downsizing if Player 1 is powerful times the probability that he is powerful. The expected payoff to downsizing is given as a function of Player 2's decision given that he observes downsizing.

The expected payoff to not downsizing is a similar weighted average of the payoffs to not downsizing if the firm is weak or powerful, expressed as a function of Player 2's decision based on observing no downsizing.

Player 1 will downsize whenever , and will refuse to downsize if . He will be indifferent if .

Now that Player 2's strategies have been specified as a function of all possible strategies of Player 1, and Player 1's strategies have been specified as a function of all possible strategies of Player 2, the equilibrium will be determined. There are three possible cases: (Player 1 never downsizes), (Player 1 randomizes), and (Player 1 always downsizes).

If Player 1 randomizes, then Player 2 will know both p and q are 0.8. Player 2 will therefore always buy, and since , the payoff to downsizing will be 2.2, and the payoff to not downsizing will be 2.8. Player 1 will therefore not be indifferent as we have assumed, and will always want to maintain the current payroll. Therefore, he will not randomize at all.

Now, assume that , and Player 1 will never downsize. Player 2 will set q=0.8, and therefore , so he will always buy if he observes downsizing. Player 2's left information set will not be reached, so he is free to specify any value of p that forces Player 1 to stick to his strategy of never downsizing. Knowing that , downsizing is only profitable to Player 1 if

This is a sub-game perfect Nash Equilibrium since Player 1's strategy is a best response to Player 2's, and Player 2's strategy is a best response to Player 1's. The strategy profile is as follows.

If q is less than 1/2, Player 2 will always try to sell if his right information set is reached. His strategy is given by . This makes Player 1's expected payoff to downsizing ; his expected payoff to not downsizing will be , so Player 1 will always downsize. Since Player 1's announced strategy does not change given Player 2's response, this is also a sub-game perfect Nash Equilibrium, specified as strategy profile 2.

There are some striking similarities between this game, and the original game presented in the last chapter. However, the complete lack of information flow has a non-trivial impact on the outcome of the game. The equilibria in which neither type of firm downsizes are similar, except for Player 2's strategy at the left node, which by Player 1's strategy choice, will never be reached in equilibrium. Likewise, in the equilibrium in which everyone downsizes, the strategies are the exact same for the imperfect information game of AT&T and for the no information game, except for the strategy choice of the investor at the nodes which the firm's strategy specifics will never be reached. For both games, the observed interaction between the two players is the same (Player 1 downsizes, Player 2 buys), but the threats of what Player 2 will do to Player 1 if Player 1 decides not to downsize are much different. Since the observed interaction doesn't change, there is no discernible impact on the equity value of the firm in equilibrium, since the strategies only change at nodes that are never reached in equilibrium.

This result is clearly not trivial given the specifications of the game. It demonstrates that investors' observed equilibrium behavior will be exactly the same regardless of how much information the management has about the firm's outlook even though their announced strategies will differ. Another important result is that even though firms do not know whether or not they are weak or strong, they will never randomize. In other words, a firm that does not know the state of the economy will never be indifferent between downsizing and not downsizing, no matter what strategy the investor happens to announce.

The outcome obtained with the game presented for AT&T was certainly a function of the management's compensation. In such a firm, the board of directors wanted to align the interests of the stockholders with those of the management, so it paid the managers well with stock options that rewarded the management with two points every time the stock price went up. Additionally, they received one point for making wise business decisions. Assume that the board is now able to create some sort of objective measurement of what a "wise" business decision is, and awards the management 2 points for making the right labor choice given the firm's economic state. The board ties compensation heavily to business performance, and not as much to the stock price in this case. Therefore, the stock options that the CEO is holding are not as important, and the management receives only 1 point if the stock appreciates in value during the time period of this game. With the revised payoffs, the game now looks as follows:

First, the management's strategy choices will be considered. The firm knows that if it is powerful, and it downsizes, that the best it can ever to is to receive 1. In this case, the investors would be buying the stock with probability 1. If the firm is powerful and maintains the labor force, the worst it could ever do would be to receive 2. In this case, the investors would always be dumping the stock. Since the worst that Player 1 can do by not downsizing is better than the best Player 1 can ever do by downsizing, no Player 1 will ever downsize, no matter what Player 2's strategy.

For a weak Player 1, the worst he could ever do by downsizing would be to have his stock sold by the investor all the time. This would mean that his stock options would be worth zero, and he would only be compensated based on making the right decision. This corresponds to a payoff of 2. If a weak Player 1 decides not to downsize, the best he could ever do would be to earn 1. In this case, the investor would always buy, and the stock options would be worth 1, but he would earn nothing else, since he made the wrong choice. Since the worst Player 1 could do by downsizing is strictly better than the best he could do by not downsizing, a weak Player 1 will always downsize.

The probability of downsizing (D) and of maintaining the workforce (M) can be calculated as the sum of weighted conditional probabilities of being powerful (P) or weak (W). The probability that there will be mass layoffs is given as

Likewise, the probability of no layoffs, P(M), is 0.8 = 1 - P(D).

Knowing this, Player 2 will update the Baysean probabilities at each of his information sets. At the left information set:

Such an investor knows that he is always dealing with a weak firm. The investor knows that he will never be at the top node if the left information set is reached. The investor will clearly always sell the stock, since the payoff from doing so (2) is larger than the payoff of buying (1). And at the right information set:

He knows he will never be at the bottom node of his information set if he is reached. This investor believes that he is always dealing with a strong firm. The investor will always buy the stock, since the payoff from acquiring the stock (2) is always larger than the payoff from divesting it (0). This strategy is clearly sub-game perfect; it is not even necessary to verify that Player 1's strategy is a best response to Player 2's, since I have proven that a weak Player 1 will always downsize and a strong Player 1 will never downsize, regardless of what Player 2 will threaten or promise to do to him. There is only one equilibrium to this game, specified as follows.

This equilibrium is clearly a separating equilibrium. Even though there is imperfect information, Player 2 knows exactly what kind of firm he is dealing with based on the firm's decision to downsize or not. In this case, downsizing is considered to be a perfect signal for a firm's health. In the case of a perfect signal, the Bayesian probability of strength or weakness is either 0 or 1, based on whether or not downsizing is observed, regardless of what the prior probabilities are. Downsizing is an unambiguous, perfect indication of firm health. This version of the game indicates that players will receive the most information when management's compensation is tied to its performance in running the company, not the stock's performance in the market. Such a result goes contrary to the popular modern notion that stock options are wonderful since they reduce the agency costs between the management and the owners, and that they align the interests of the two. But the AT&T game proves that this is not true: management too obsessed with the stock's performance are willing to make bad decisions if they think the market will boost the stock price, regardless of whatever consequences the decision eventually has for the firm's business. In this case, the management is somewhat insulated from the mood swings of the stock market, and is paid only to do the right thing. Ironically, it is when stockholders offer the management fewer stock options that they get the most information out of the management as to the health of the firm.

The difficulty is how to compensate management for making the "right choice"-especially when such decisions are highly subjective. Since decisions are made under uncertainty, it remains complex to determine whether or not management did the "right thing." A CEO could have fortuitously stumbled into a good decision, and should the shareholders compensate him for being in the right place at the right time? Or what if a CEO had carefully considered an array of alternatives, and chosen the best one, which for reasons beyond his control, turned out to be a bad choice. Should a CEO not be compensated as well in such a scenario? The compensation that a CEO would receive becomes much riskier the more his salary will fluctuate with profits. In terms of risk-sharing between the investors and their hired management, it might not be a good idea to make the CEO's compensation more risky. It could be costly for the investors to ask the CEO to bear more risk, since he might make strategy choices that minimize the fluctuations in his own compensation, and not the choices which maximize firm value.

In the circumstances of this game, the CEO knows with perfect certainty the future health of the firm, and is then asked to make a decision. The salary to managers should be structured as it appears in this game, if and only if managers have perfect information of the firm's health, and there exists an objective, undebatable way to measure whether or not the CEO has made the right choice.

By altering the basic game presented in the previous chapter, these games were able to capture the change in the strategic interactions between investors and firms given different circumstances regarding information and incentives to the managers. Perfect information was found to benefit the investors substantially. Investors could never do better in the original game than they could in the game of perfect information. A game of no information (not even the firm knows its own health) changed the strategy profiles of each player, but only the strategies of the investor that would never be executed, given the equilibrium strategies of the firms. In this game, the observed interaction between the players does not change. The outcome of the game suggests the value of the firm is not only a function of its earnings potential, the maintenance of its market share, and the competency of its management, but also a function of the information flow between the firm's owners and its agents.

Changing the incentives to management also substantially alters the equilibria. Many economists currently believe that stock options are a wonderful idea in that they alleviate the agency costs of running a firm, since they align the interests of management and owners. Yet this analysis proves that stock options might actually increase agency costs (and decrease the value of the firm) if management becomes too obsessed with boosting the current stock price, and not with building the actual value of the firm. The perceived value of the firm (the stock price) can diverge from the actual value of the firm in the presence of information asymmetries. Stock options make potentially harmful moves like downsizing more attractive to management. This is observed in sub-game perfect equilibria, in which strong firms downsize, even though it is a bad business strategy.

Such an incentive seemed especially evident in the results obtained in Chapter III, in which firms that downsize unexpectedly (or firms that fire more workers than the market thinks they should) find their stock appreciating on the news of downsizing. Stock options and asymmetric information make for a dangerous combination if the management has an incentive to take actions that raise the perceived value of the firm with the side effect of lowering the actual value of the firm. This does not indicate that investors are stupid or irrational when valuing the firm, but the perceived value of the firm and the actual value of the firm can differ if the investors are not informed as to the state of the firm's health.

Chapter VII addresses the case of Delta Airlines, in which management downsized a substantial portion of the workers to boost the stock price, only to have the move completely backfire, and cause Delta's stock to fall behind its competitors' equity. It will also discuss alternatives to downsizing, such as those followed by United Airlines. The next chapter, Chapter VI, assesses the validity of the equilibria given Nature's choices for strength and weakness, and offers a further extension to the game by introducing a third type of firm: one which is inefficient and can't downsize. This third type will be useful in explaining the corporate strategy of Delta and United in Chapter VII.