One never, of course, does such things for one's own benefit. Rather, they are done to influence others playing the game to alter their course of action or reaction to a more favorable path. As such, these negative actions, the dismantling of a plant or the entering into of a binding contract removing a strategic possibility, must be done publicly. It must be observed and understood by others playing the game to have any effect.
One of the most famous situations illustrating the folly of commitment in private appears in the movie Dr. Strangelove where, in the climactic scene, it is revealed that the Russians have built a doomsday machine set to go off in the event of nuclear attack. Moreover, to complete the commitment, the Soviets have added a self-destruct mechanism to the device. It also goes off if tampered with or turned off. Since the machine will destroy the Earth, it ought properly to dissuade all countries to engage in nuclear combat.
But there's a problem--the US is only made aware of the machine after non-recallable bombers have been launched to deliver a devastating nuclear attack at the behest of a berserk air force commander. Why did the Soviets not tell the US and the world about the machine, asks Doctor Strangelove to the Soviet ambassador?
The premiere likes surprises.comes the pitiful answer. And so unobserved commitment is, in effect, no commitment at all.
While paradoxical initially, the idea that fewer options can improve one's strategic position is intuitive once grasped and was understood at an intuitive level long before the invention of game theory.
But I want to talk about a less well-known paradox: if such commitment strategies succeed by altering others' play in a more favorable direction from the perspective of the committing party, why would these others choose to observe the commitment in the first place? Shouldn't they commit not to observe in an effort to frustrate the commitment efforts of others? It turns out that this second level of commitment is unnecessary, at least in the formal argument, all that is needed is a small cost to observe the choices made by the committing party for the value of commitment to be nullified.
For example, two players are playing a WEGO game where they choose between two strategies, S and C (labels will become clear shortly). The equilibrium in this game turns out to be (C, C), but player 1 would gain an advantage if she could commit to strategy S, which would provoke S in response, and raise her payoff. Thus, strategy C can be thought of as the Cournot strategy while S represents the Stackelberg strategy in terms of archetypal quantity setting games. Suppose further that, if 1 could be sure that 2 played S, she would prefer to play C in response, so (S, S) is not an equilibrium in the WEGO game.
The usual way player 1 might go about achieving the necessary commitment is by moving first and choosing S. Player 2 moves second, chooses S in response, and lo and behold, IGOUGO beats WEGO as per our theorem. Player 2 is perhaps worse off, but player 1 has improved her lot by committing to S.
But now let us slightly complicate the game by paying attention not just to the transmission of information but also its receipt. After player 1 moves, player 2 can choose to pay an observation cost, c, to observe 1's choice perfectly. This cost is very small but, if not paid, nothing about 1's choice is revealed to 2. After deciding on whether to observe or not, player 2 then chooses between C and S and payoffs are determined.
Looking forward and reasoning back, consider the situation where 2 chooses to observe 1's move. In that case, she chooses S if player 1 chose S and C if player 1 chose C. So far, so good. If she does not observe, then she must guess what player 1 might have chosen. If she's sufficiently confident that player 1 has chosen S, then S is again the best choice. Otherwise, C is best.
So should she observe or not? If commitment is successful, then player 2 will anticipate that player 1 has chosen S. Knowing this, there is no point in observing since, in equilibrium, player 2 will choose the same action, S, regardless of whether she looks or not. Thus, the value of information is zero while the cost gathering and interpreting the information is not, so, behaving optimally, player 2 never observes and thereby economizes (a little) by avoiding the cost c.
But then what should player 1 do? Anticipating that player 2 won't bother to observe her action, there is now no point in playing S since C was always the better choice. Thus, player 1 will choose C and it is now clear that the whole commitment posture was, in fact, mere stagecraft by player 1.
Of course, player 2 is no fool and will anticipate player 1's deception; therefore, if the equilibrium involves no observation, player 1 must have chosen C, and hence player 2 chooses C. Since we know that player 2 never pays the (wasteful) observation cost in equilibrium, the only equilibrium is (C, C), precisely as it was in the WEGO game. In other words, so long as there is any friction to observing player 1's choice, i.e. to receiving the information, first-mover commitment is impossible.
The issue would seem to be the conflict between players 1 and 2 where the latter has every incentive to frustrate the commitment efforts of the former since, if successful, 2 is worse off. But consider this game: Suppose that (S, S) yields each player a payoff of 3. (C, C), on the other hand, yields each player only 2. If player 2 chooses C in response to player 1's choice of S, both players earn zero while if the reverse occurs, player 2 chooses S and player 1 chooses C, then player 1 earns 4 while player 2 only earns 1. This fits our game above: C is a dominant strategy for player 1 while 2 prefers to match whatever 1 does.
This game has some of the flavor of a prisoner's dilemma. It is a one-sided trust game. By playing the WEGO game, both players lose out compared to the socially optimal (S, S), yet (S, S) is unsustainable because 1 will wish to cheat on any deal by selecting C. One-sidedness arises from that fact that, while player 1 can never be trusted to play S on her own initiative, player 2 can be trusted so long as he is confident about 1's choice of S.
Player 1 seeks to overcome her character flaw by moving first and committing to choose S, anticipating that 2 will follow suit. Surely now 2 will bother to observe if the costs are sufficiently low? Unfortunately, he will not. Under the virtuous (S, S) putative equilibrium, player 2 still has no reason to pay to observe player 1's anticipated first move since, again, 2 will choose the same action, S, regardless. Knowing this, 1 cannot resist the temptation to cheat and again we are back to (C, C) for the same reasons as above. Here the problem is that, to overcome temptation, 1 must be held to account by an observant player 2. But 2 sees no point in paying a cost merely to confirm what he already knows, so observation is impossible.
What is needed is a sort of double commitment--2 must first commit to observe, perhaps by prepaying the observation cost or by some other device. Only then can 1 commit to play S, and things play out nicely.
While paradoxical and logically correct, it seems quite silly to conclude that effective commitment is impossible. After all, people and firms do commit in various ways, their choices are observed, and these commitments have their intended effect. So what gives?
One answer is that, in reality-land, strategies are not so stark as simply S or C. There are many versions of S and likewise of C and the particular version chosen might depend on myriad environmental factors not directly observable to player 2. Now the information may be valuable enough that observation is optimal.
Seeds of doubt about the other's rationality can also fix the commitment problem. Ironically, this cure involves envisaging the possibility of a pathologically evil version of player 1. These evil types always act opportunistically by choosing C. Now there is a reason for 2 to look since she cannot be certain of 1's virtue.
A third possibility is that observing is simply unavoidable or that observation costs are negative. Curiosity is a trait common to many animals including humans. We derive joy from learning new things even if there is no direct economic value associated with this learning. Thus, individuals pay good money for intro texts on literary theory even though, for most of us, learning about Derrida's theories of literary deconstruction is of dubious economic value. Obviously, if the cost c were negative, i.e. a benefit, the problem vanishes and commitment is restored.
So if the theory is so implausible, then why bother bringing it up? One answer is to point out some countermeasures to commitment strategies. After all, if player 1 can "change the game" by committing to a strategy first, why can't player 2 change the game by committing to be on a boat in Tahoe and hence out of touch with what 1 is up to? A better answer is that it highlights the fact that commitment is a two-way street. Effective commitment requires not just that player 1 transmit the commitment information but that player 2 receive (and correctly decode) this information. Game theorists and others have spent endless hours thinking up different strategies for creating transmittable information, but precious little time thinking about its receipt. My own view is that this is a mistake since deafness on the part of other players destroys the value of commitment just as effectively as muteness on the part of the committing party.
Returning to Strangelove, it's not enough that the Soviet premiere transmit the information about the doomsday device ahead of time, for commitment to be effective such information must be heard and believed. This suggests the following alternative problem--even if the premiere had disclosed the existence of the doomsday machine, would the US have believed it? If not, Slim Pickens might still be waving his cowboy hat while sitting atop a nuclear bomb plummeting down to end all life. Yee-hah!