Consider the equilibria of the AT&T game in which all types of firms downsize. The Bayesian probabilities that Player 2 is facing a strong or weak Player 1 at his left information set are simply the prior probabilities, because Player 1 downsizes with probability 1. Based on these probabilities, Player 2 will always buy the stock, since his expected payoff, a function of these prior probabilities, is higher for buying the stock than it is for selling the stock. The expected payoff to selling the stock is

The utility is clearly a decreasing function of p; as p increases, selling the stock becomes a less rewarding option.

Similarly, as p decreases, selling the stock becomes more and more attractive in terms of expected utility. For large values of p (such as 0.8, in the AT&T game), the payoff to buying is always higher. The value of p that makes the investor indifferent between buying and selling is the value of p that that sets the payoffs to each option equal to each other:

This result can also be obtained algebraically, abstracting from the numeric utility values that I've assigned. Without any loss to generality, the payoff to dumping a weak stock will be given as X, and the payoff to buying a strong stock will be Y. A player will receive Z if he is holding the stock of a company that demonstrates a good business strategy (such as downsizing in a weak market). Investors will receive 0 for selling the stock of a strong company. X, Y, and Z are all positive quantities, the only other restriction is that X > Z; else, the payoff to buying will always exceed the payoff to selling, regardless of whether or not the firm is strong or weak. The algebraic solution of indifference, where p is the threshold probability (the probability that makes the payoff to buying equal to the payoff to selling) is given by

The threshold probability is an increasing function of X. As X rises, the equilibria of chapter IV become less common, as they only hold for a more limited range of high values of p.

The threshold probability is a decreasing function of both Y and Z. As Y or Z fall, the equilibria of Chapter IV will hold for a wider range of values of p. Recalling that X > Z always, and that the denominator is positive for all possible values of X, Y, and Z:

Until now, we have always assumed that the game is played during a time of economic prosperity. In this variation to the game, Nature has determined that there will be a recession. The probability of weakness is 60%, and the probability of strength is 40%. This is a specific example of the case in which the probability of weakness is actually higher than the probability of strength. Simply put, the probability of strength is below the threshold probability. If all firms downsize, or if all firms maintain, then the investor will know that it is more likely that he is dealing with a weak firm, so he will always sell. This will cause the original equilibria to collapse, and new equilibria will have to be computed.

As a starting point for the analysis, assume that all firms want to make the right decision. Strong firms want to maintain the labor force, and weak forms want to downsize. Player 2 knows that every time his left information set is visited, he is dealing with a weak player so he will sell the stock. If Player 2's right information set is reached, he is dealing with a powerful firm, and will buy the stock. Knowing Player 2's strategy, a weak Player 1 will get 1 for downsizing (his stock is sold by Player 2), and 2 for not downsizing (his stock is bought by Player 2). Therefore, this is not an equilibrium, since a weak Player 1 will want to deviate from his strategy and refuse to downsize.

Will a weak Player 1 agree to randomize if a strong Player 1 never downsizes? For a weak player to randomize, the payoff to downsizing and not downsizing must be the same for him to be indifferent between his strategies. This is given as

The only kind of player ever downsizing is weak, so if Player 2's left information set is reached, he knows that he is dealing with a weak firm, since and . The prior probability of downsizing is therefore given as

The probability of Player 1 maintaining the workforce is therefore .

There exists a unique probability of a weak player downsizing that will allow q = 1/2, the condition necessary for Player 2 to randomize at his right information set.

The overall strategy profile is given as

This is the unique sub-game perfect Nash Equilibrium of this game. Player 1's payoff to downsizing if he is strong is never as good as his payoff to not downsizing, so he will always maintain the payroll. A weak Player 1 will do just as well by downsizing or not downsizing, so he is happy to randomize according to . If Player 2's left information set is reached, he knows he is addressing a weak firm, so he will always downsize. If Player 2 observes no downsizing, he is indifferent between buying and selling, and is able to randomize between these two equally-profitable strategies according to . Since Player 1's strategy is a best response to Player 2's and vice-versa, this strategy Profile is a sub-game perfect Nash Equilibrium.

There are no cases in which all players will maintain. If no player downsizes, then Player 2's right information set will always be reached. Given that there are more weak firms than strong firms, Player 2 will always sell stock. The worst a weak Player 1 can do by downsizing is always better than not downsizing and being sold off, so a weak Player 1 will want to deviate form the announced strategy and downsize. This is clearly not an equilibrium.

There are no equilibria in which a strong player will ever downsize. If all players downsize, then Player 2's left information set will always be reached with probability 1. Since Player 2 will be confronting weak firms more often than strong firms, the stock will be sold. Downsizing and getting sold off is unequivocally worse for a strong firm than any result from maintaining the work force, so a strong Player 1 won't want to downsize any longer. A weak Player 1 will never want to randomize if a strong Player 1 is always downsizing. In such circumstances, if an investor observes no downsizing, he will conclude that he is always dealing with a weak firm, and he will sell the stock. A weak player 1 can always do better by downsizing, and won't want to randomize.

There are also no equilibria in which a strong player will randomize. To get a strong Player 1 to randomize, the payoffs to downsizing and not downsizing must be the same. As earlier, this condition is given by . If strong firms randomize and weak firms downsize, then the only firms that won't downsize are strong. The investors know that every time they see a firm maintain the labor force, that it is strong. Investors will set . There now exists no value of to set , which is the required condition. Will weak firms maintain the labor force if strong firms randomize? The only types of firms downsizing are strong, so investors will always buy the stock of downsizers. Downsizing will now be attractive to weak firms, since they not only will be bought, but also get an extra point for making the right decision. Weak firms will never downsize if strong firms are randomizing over strategies. It can also never be the case that both strong and weak firms are randomizing. The condition for a strong firm to be indifferent between his strategies is , and for a weak firm it is . This quantity cannot both equal 1 and -1 at the same time, so this strategy profile can never occur in equilibrium.

All candidates for equilibria have been proven to be unacceptable except for the strategy profile in which strong firms never downsize, weak firms randomize, investors observing downsizing sell the stock, and investors observing no downsizing randomize. This is not a trivial result from changing Nature's choices of which firms will be strong or weak. There exists only one equilibrium, so in a recession, strong firms will never downsize, and weak firms will downsize twice as less often as they don't downsize. In a recession, investors who observe downsizing know that they are dealing with a weak firm, so they will sell the stock. They will sell and buy the stock of firms that don't downsize with equal probability. Perhaps this explains why the stocks of strong firms get dumped in a recession, since they may be mistaken for weak firms. Weak firms don't downsize all of the time, since they can attempt to masquerade as strong by maintaining the labor force, and will be bought 50% of the time.

This equilibrium is partially separating, since downsizing is a dead give-away that a firm is weak, while not downsizing leaves investors confused between who is strong and weak. The solutions of the game in a boom (Chapter IV) and in a recession demonstrate the bizarre observation that strong firms lay off workers only in a boom, but never in a recession. The payoffs and structure remain the same, but the prior probabilities of economic well-being clearly impact the outcome. When the probability of weakness rises, investors become worried that they are dealing with a weak firm. In the only equilibrium, any firm displaying weak behavior (downsizing) is immediately sold. Strong firms need to convince investors that they're not weak, during a recession in which the majority of firms are weak. Therefore, they always avoid downsizing, a luxury which weak firms simply don't have. In a boom, strong firms don't mind downsizing if weak firms are also downsizing. Investors know that the probability that any given firm is strong is sufficiently high, so if all firms downsize, any firm observed downsizing is probably strong, and is therefore bought.

Such an equilibrium can be applied to the recession of the early 1990s, in which strong firms were not downsizing, and some weak firms were. The phenomenon of strong firms downsizing is only observed after the recession ended. The games suggest that during recessions, strong firms will never downsize, but during an economic boom, there exists an equilibrium in which strong firms will always downsize. Although it appears counter-intuitive, it is supported not only in the dynamics of these games, but is also anecdotally observable in the stock market. Perfect examples are companies like AT&T and Procter & Gamble, which did not lay off a single worker during the recession, but once they regained profitability-and record profitability at that-they announced plans to eliminate a substantial percentage of their workforce.

The final extension to the game is very similar to the model developed with AT&T, except this time, there are three kinds of firms: firms that will be strong in their market, firms that will be weak, and firms that are horribly inefficient. The strong and weak firms have the choice of downsizing or maintaining their labor forces, while the inefficient firms are so poorly run, that they cannot spare to loose any employees or the entire operation will collapse. Assume that they are producing a product in series, and if any one worker is not on the assembly line, no products can be produced, since all output must pass though that worker. Management is so inept or so ignorant of market conditions, that it has neither the willingness, resolve, nor motivation to fire any employees. The company is more or less on auto-pilot, and there is no strategy choice, so the firm just maintains its current level of employment.

Nature determines who is strong, weak, or inefficient. The payoffs will be the same as always, with additional utility values specified for the management and investors of an inefficient firm. The shareholders love to see the stock appreciate, so the board of directors has included hefty stock options in the executive's compensation. The executives will be given two points if the stock appreciates. If they make the correct business move, they will get an extra point (perhaps the board sees they made a wise choice and increases their bonuses). Doing the right thing is defined as either maintaining employment in a strong state, or downsizing in a weak state. Losing one point for their lack of foresight and lack of managerial skill will punish inefficient executives. So if the firm is inefficient, and the stock is dumped, management will have a utility of -1. If the stock is bought, their utility will be minus one point for being poor management (they cant help it-nature made them that way!) and plus two points from the stock options for having their stock bought; their overall utility is 1 in this case.

Investors don't know whether a firm is strong, weak, or inefficient. Investors will receive 2 points for selling a stock that turns out to be weak, and receive no points for selling a stock that proves to be strong. They will receive one point for buying stock that turns out to be strong, and no points for buying a stock that is revealed to be a weak company. The stakes are much higher for a firm that is inefficient, and probably headed to bankruptcy. The investors will receive five points for dumping a stock that winds up to be inefficient, and will lose five points for buying an inefficient stock, since their investment will most likely be wiped out.

The addition of a third style of firm might be a better way to classify the airline industry than a simpler two-state game. Airlines could be winners or losers, or they could be so horrendous that they completely wipe out their net worth altogether, like Continental, Pan Am, or Eastern in the early 1990s. Winner airlines will increase their market shares and attract new customers, losers will be less profitable than desired, and inefficient firms will be worth more dead than alive.

Over the short run, this might not be a bad way to think of corporations. With clever accounting, many companies can beef up accounts receivable, or play with extraordinary charges, and dupe the auditors and shareholders for a few accounting periods before the accounting irregularities catch up with them. Also, many shareholders have no idea who the firm's twenty senior executives are, or what their intelligence and track record are. It's very possible for a firm to be either strong, weak, or inefficient without anyone knowing (over the short run) to which category it belongs. Once the employment decisions are made, and the shareholders place their bets, the actual state of nature is revealed, and the payoffs are given.

As in the previous game, Nature moves first, and selects the prospects for the firm and intelligence of its managers. Then, intelligent firms are allowed to choose between downsize and maintain, while inefficient firms must choose maintain. Payoffs are given as ordered pairs, with the first utility number being the firm's payoff, and the second being that of the investors.

There are three Sub-Game Perfect Nash Equilibria. In the first, no firm will downsize.

Player 1's strategy specifies that Player 2's right information set will be reached with probability 100%. When Player 2 is reached, he uses Bayes' Law to update the probability of which node he is at. This is done as follows.

The expected payoff to selling the stock is

The expected payoff to buying is

Player 2 is therefore indifferent between buying and selling when all three types of firms never downsize. To ensure that a strong Player 1 will never downsize, the expected payoff to maintaining the labor force must be greater than the expected payoff of downsizing. This is achieved when

Likewise, to ensure that a weak Player 1 will never randomize:

This becomes a more stringent inequality than the condition for a strong Player 1, since it can be re-written as

The Bayesian probabilities of being reached are meaningless. Modern game theory allows us to specify any set of beliefs that justify Player 2's decision to sell the stock of all downsizing firms.

The payoff to selling the stock is

The payoff to buying the stock is

The complement to this strategy is one in which all strong and weak firms downsize, and only the inefficient firms are left with the same amount of employees by the end of the game. The investors know for sure when a firm is inefficient, so they punish it by dumping its stock.

Now, the probability of being of a certain type of firm and the probability of making a certain decision are no longer independent. We are able to ascertain what percentage of firms overall will be downsizing based on the strategies, since we know a priori the percentages of firms of each type, and what strategy each type will pursue. Using the notation for strategies presented in the game tree, the prior probability of whether firms are downsizing is given as:

Likewise

Using Bayes' Law, the investors can ascertain the probability that they are at a given node, given that their information set has been reached. If Player 2 observes that the firm is not downsizing, he knows with 100% certainty that he is dealing with an inefficient firm.

His payoff to buying stock is -5, and 5 to selling. Therefore, Player 2 will sell the stock of all firms he observes maintaining their payrolls. If Player 2 observes downsizing, he once again uses Bayes' Law to determine the probability that he is at a certain node in his information set.

The payoff to Player 2 of buying the stock is

and the payoff to selling is

so Player 2 will always buy. Having determined what Player 2's best responses are given Player 1's strategy, it must be the case that Player 1 now has no profitable deviation from his strategy in order for the given strategy profile to be a Nash Equilibrium in the sub-game perfect sense. If a strong type of Player 1 downsizes, he will receive 2, since the investors will buy his stock. If he does not downsize, he will receive 1 since investors will sell his stock. Therefore, Player 1 will downsize if he is strong. If a weak Player 1 downsizes, he will receive 3 since investors will buy his stock, and if he does not downsize, he will receive 0 since investors will sell his stock. A weak Player 1 will downsize, so all firms (except inefficient ones) will always downsize. Since Player 1's strategy is a best response to Player 2's, and vice-versa, this strategy profile entails a sub-game perfect Nash Equilibrium.

This game serves as a perfect example of why the market adored downsizing in the 1990s: inaction on the part of management to slash jobs was viewed as a sign of complacency at best, or at worst, being completely out of touch with the current economic reality, even though that "reality" was an economic boom. All companies that maintained their workforces were considered "inefficient" and were dumped, while strong and weak firms downsized, and were rewarded. The managers at inefficient firms saw how they were getting beaten up by investors while both strong and weak firms were appreciating in value, and there remained nothing that they could do about it, since there was nowhere for them to cut without the entire operation collapsing.

The game has a third equilibrium in which a strong firm is made indifferent between downsizing and not downsizing. At first, it seems absurd that a firm would not care one way or another what its work force will be. Yet, if the management envisions the same payoffs regardless of the strategy the firm takes, then the firm would be willing to randomize, since any combination of two strategies offering the same level of utility will offer the exact level of utility that each strategy alone provides. In such a case, investors announce a strategy that makes firms not care one way or the other. Before, investors were willing to randomize over buying and selling since they did not know what kind of firm they would be dealing with; now firms are willing to randomize since the investors will randomize between buying and selling. The strategy profile is as follows:

First, I will calculate the probabilities of downsizing and not downsizing, conditional on firm types:

The probability of maintaining the workforce will therefore be given by

A strong type will randomize only if he observes that he will do just as well regardless of whether he downsizes. This is given by

Assume for the time being that investors will always sell off firms that don't cut their payrolls: . This is the worst possible scenario for a strong Player 1, since it minimizes his payoff if he decides not to downsize. This changes the condition to

So when investors always dump stocks of non-downsized companies, and dump stocks of downsized companies exactly half of the time, a strong firm will be indifferent between downsizing and not downsizing. This implies two restrictions which must be verified: Player 2 prefers selling the stocks of non-downsized companies, and Player 2 is indifferent between buying and selling stocks of downsized companies (otherwise, he would not randomize according to ).

For Player 2 to be indifferent, the payoff to selling must be equal to the payoff to buying.

This determines what the probability of downsizing given strength must be in order to induce an investor to randomize should he observe downsizing. Assuming for now that all weak players always downsize,

So a strong Player 1 must downsize exactly 2/7 of the time, and not downsize exactly 5/7 of the time to induce Player 2 to randomize at his left information set. We have required Player 2 to always sell if he observes no downsizing, and must now verify that Player 1's actions make this the case. We have already required that and that , so Player 2 can update the probabilities that he is at certain nodes in his right information set as follows:

Solving for q using and ,

Likewise, . So when Player 2 observes no downsizing, he concludes that he is dealing with a strong firm five-sixths of the time, and an inefficient firm one-sixth of the time. The payoff to buying is given by

The payoff to selling is

So, Player 2 is indifferent between buying and selling at his right information set. This allows him to make the choice of , which means he is always buying. In actuality, we have made Player 2 indifferent at both of his information sets given Player 1's strategy. So one of the infinitely many best responses to Player 1's strategy is to buy and sell with equal probability if downsizing is observed, and to always sell if it is not observed. Given this choice as Player 2's strategy, it must be proven that Player 1 will not deviate from his announced strategy for this profile to be a Nash equilibrium.

Given Player 2's announcement, a strong Player 1's payoff to downsizing will be if he downsizes since Player 2 randomizes 50%/50%, and 1 if he does not (Player 2 sells the stock). So Player 1 is indifferent, and is happy downsizing 2/7 of the time. A weak Player 1's payoff to downsizing is , and 0 if he does not, so he will always downsize. Since Player 1's strategy is a best response to Player 2's, and vice-versa, this strategy profile entails a sub-game perfect Nash Equilibrium.

These are the only possible equilibria for this game. It will never be the case that both strong firms and weak firms will randomize. For a strong firm to randomize, we would need

And for a weak firm

But the quantity to the right can never equal both 1 and -1, so this is impossible. Only one type of firm can randomize at a time. Could it ever be the case that strong firms maintain the work force and weak firms randomize? In such a scenario, the only kind of firm to downsize will be weak, so investors will always know they are dealing with a weak firm when they observe downsizing, so they will sell the stock. Weak firms would much rather be bought, so they won't downsize at all, and will always maintain (Strategy Profile 1). Will a weak firm randomize if a strong firm always downsizes? For a strong form to downsize, the payoff to doing so must be greater than the payoff to maintaining the labor force. This condition is given from above as . But since we require that for a weak Player 1 to downsize, there are no strategies in Player 2's set that could ever allow this to happen; this quantity cannot be both equal to -1 and greater than 1 at the same time!

Also, there are no equilibria in which strong players always downsize, while weak players always maintain; moreover, there are no equilibria in which strong players always maintain and weak players always downsize. In the first case, if Player 2's left information set is reached, he will know for sure that he is dealing with a strong firm, and therefore buy the stock. Knowing that firms who downsize will have their stocks purchased, a weak Player 1 will deviate from his strategy of maintaining the labor force, and will start to downsize. As for the second equilibrium, the only players observed downsizing will be weak, so Player 2 will always sell stock if he observes downsizing. If Player 2's right information set is reached, he knows that the probability is very high that he is dealing with a strong firm (as opposed to an inefficient firm, since in this case, weak firms always downsize), so he will buy the stock. Knowing this, no weak Player 1 will want to downsize anymore if Player 2's strategy is to dump downsizers and accumulate shares in non-downsizers. Since this exhausts all possible strategy combinations, only the three profiles presented above constitute equilibria.

This game will be useful to describe the strategy choices of Delta Airlines and United Airlines, which both contemplated major downsizings in the mid 1990s. The sub-optimality of such a game is that strong firms, which should never downsize if information were prefect, are observed downsizing, simply to convince the market that they are not inefficient. This was the actual behavior of Delta, which went to great lengths to prove its efficiency, but at the cost of losing valuable members of its workforce.