Thursday, September 18, 2014

Secrets and Lies

There are many instances where players in a game can control the timing and transparency of their moves. Indeed, a fundamental question firms face when making key strategic decisions is whether to keep them a secret or to reveal. Geographic decisions like plant openings or closings, strategic alliances with foreign partners, or overseas initiatives are often revealed, sometimes with great fanfare. Other decisions, such as the details of a product design or merger target, are closely guarded secrets. Firms also tell lies (or at least exaggerations) at times, for instance, in the schedule for a software release or in plans to acquire targets done to jack up its price to a competitor. What can game theory tell us about secrets, lies, and timing.

One way of thinking about secrets is to imagine a game whose timing is fixed but where disclosure is at the discretion of the participants. For instance, firm 1 moves first, followed by firm 2. When firm 1 moves, it can choose to (truthfully) disclose its action to the world or not. This amounts to the choice between a WEGO game versus an IGOUGO game from the perspective of firm 1. The key question then, is when should firm 1 disclose its strategy. For a large class of situations, game theory offers a sharp answer to this question:

Disclosure is always better than secrecy.

On to the specifics: Suppose that both firms are playing a game with finite strategies and payoffs and complete information; that is, both firms know precisely the game that they are playing. Let x denote the firm 1's choice and y denote firm 2's. Let y(x) denote how firm 2 would respond if it thought firm 1 were choosing strategy x. Let x(y) be similarly defined for firm 1. A pure strategy equilibrium is a pair (x', y') where  y' = y(x') and x'=x(y'). Let P(x, y) be firm 1's payoff when x and y are selected. Thus, in the above equilibrium, firm 1 earns P(x', y') or, equivalently, P(x', y(x')).

Now consider an IGOUGO situation. Here, firm 1 can choose from all possible x. Using look forward, reason back (LFRB), firm 1 anticipates that 2 will play y(x) when 1 plays x. Thus, 1's payoffs in this situation are P(x, y(x)). Notice that, by simply choosing x = x', 1 earns exactly the same payoff as in the WEGO game. OTOH, firm 1 typically will have some other choice x* that produces even higher payoffs P(x*,y(x*)). Thus, IGOUGO is generically better than WEGO and certainly never worse.

But this begs the question of secrets and lies. Why doesn't firm 1 simply try to persuade firm 2 that it will play x* and thereby induce y* =y(x*), thus replicating the ideal setting of the IGOUGO game? One reason is that, generically  x(y*), i.e., 1's best response to 2's playing y*, is not to play x*. One might view this as an opportunity since this means that available to firm 1 is some strategy x** = x(y*) with the property P(x**,y*) > P(x*,y*). That is, if 1 can persuade 2 to play y*, it can do even better than playing x*, the IGOUGO strategy.

The problem, of course, is that firm 2 will only play y* if it is convinced 1 will play x*. Since 1 will never play x* if this persuasion is successful, then 1 can never convince 2 of its intentions to play x*. As a result, the whole situation unravels to the original WEGO outcome of (x', y'), which is worse for firm 1 than IGOUGO. Put differently, firm 1's promises to play x* are simply never credible.

This story about the unraveling of non-credible pronouncements, however, has a profound and undesirable implication. It implies that, in game theory land, it is impossible to trick or deceive the other party (at least in our class of games). Deceptions will always be seen through and hence rendered ineffective. But we see all sorts of attempts at deception in business and in life. Surely people would not spend so much time and effort constructing deceptions if they never worked. Here again, full rationality is the culprit. Our game theory land firms are hard headed realists. They never naively took the other's words at face value. Instead, they viewed them through the cynical lens that implied that promises which were not in 1's self-interest would never be honored.

Life is sometimes like this, but thankfully not always. People do keep their word even when keeping it is not in their self-interest. Moreover, others believe these "honey words," acting as though they are true, even if they are possibly not credible when put to the test. Moreover, we teach our children precisely this sort of rationality---to honor promises made, even if we don't want to. As usual, game theory can accommodate this sort of thing simply by amending the game to allow a fraction of firm 2's to be naive, believing firm 1's promises and to allow a fraction of "honorable" firm 1's who keep promises made, even when better off not doing so. Once we admit this possibility, bluffing becomes an acceptable, and occasionally profitable, strategy. A deceitful and selfish player 1 may well spend time trying to convince 2 that he will play x* and therefore 2 should play y* since there is a chance that 2 will act upon this.

This does, however, muddy our result a bit. We now need to adjust this to:

When rivals are sophisticated, disclosure is always best.

With enough naifs in the firm 2 population, 1 may well prefer to take its chances on trickery and deceit rather than committing to the ex post unappealing action x*