By Andrew Kydd
The downing of a Malaysian airliner over eastern Ukraine with the loss of almost 300 lives may open a new chapter in the long simmering conflict. It was quickly confirmed that the aircraft had been shot down, and given the cruising altitude of commercial jets, it must have been by a surface to air missile rather than small arms fire. This reduces the number of actors who could conceivably have done it and may facilitate identification of the culprits. However, we are probably in for a lot of finger pointing and vigorous denial, as the parties try to divert blame to the other side. Avoiding blame is important because the parties think the international community may impose some punishment on whichever side it decides was responsible for the crime.
Unfortunately, the fact that the international community may punish the party it deems guilty but cannot be sure who is guilty generates incentives not just to avoid blame, but to commit the crime in the first place. There is an interesting strategic dynamic at work. For instance, if the Ukrainian government thinks the rebels will be blamed for the crime, the government should commit it, since this will result in the rebels being punished. The rebels, in turn, should definitely not commit the crime because that will just bring down punishment on their own head. However, this should make the international community blame the government, since they are certain to commit the crime and the rebels are certain not to. The same holds in reverse; if the government will be blamed, the rebels should commit the crime and the government should not, but this should make the international community blame the rebels. So it is clearly unstable for one side to commit the crime and the other side not to, since this will result in the side committing the crime being punished. Should they both commit it? Should neither one? Should they flip a coin? Situations like this call for a little game theoretic investigation, which I outline below.
The main implication of the analysis is that if the international community is biased in favor of one actor, that actor is more likely in equilibrium to commit the crime than the other side is. In the Ukrainian case, if the international community is biased in favor of the government side, that actually gives the government side a greater incentive to commit such crimes in hopes that the rebels will be punished for them. Thus the presence of the external actor and its bias generates perverse incentives for actors to commit crimes that they would otherwise have no incentive to commit.
This kind of analysis is of course no substitute for actual investigation of who is responsible for the downing of the plane. In an alternative model in which blame can be allocated accurately, there is no incentive to commit provocations in hopes that the other side will be blamed. The analysis does suggest, however, that in environments where it is difficult or impossible to allocate blame effectively, the possibility of punishment does generate incentives to engage in provocations for all sides, especially for those likely to be spared by the punisher.
Consider two actors involved in a conflict and a third party observer. One of the two players is selected at random and, unbeknownst to the third party, presented with an opportunity to commit some crime that will be impossible to trace for sure to either of the two actors. If the player does not commit the crime, the game ends and no one is punished. If it does, then the third party has an opportunity to punish one or the other actor.
Payoffs are as follows. Each player gets 0 if no one is punished, 1 if the other side is punished and -1 if it is punished. The third party gets 1 if it punishes the guilty party and -1 if it punishes an innocent party. In addition, it gets a benefit b (assumed to be less than 1) for punishing player 2, reflecting a bias against that side.
After observing the crime, the third party will have some probability estimate, call it g1, that the first actor is guilty and a corresponding probability estimate g2 = 1 – g1 that the second actor did it. Call the equilibrium probabilities that the players would commit the crime if given the chance p1 and p2. If we posit that each player is equally likely to be given the opportunity, the posterior probability that player 1 committed the crime is, from Bayes Rule, g1 = p1/(p1+p2), and for player 2 it is g2 = p2/(p1+p2). So the third party’s beliefs about who committed the crime depends on their actual likelihood of committing it in a straightforward way, the greater p1, the greater g1 and likewise for p2 and g2.
What are the equilibrium likelihoods that each side will commit the crime? If we try to imagine an equilibrium in which one side is punished for sure, we know the other side must commit the crime for sure if it has the chance and the side getting punished will not commit the crime. This will invalidate the third party’s strategy because now the side being punished has zero posterior likelihood of having committed the crime.
Therefore, the only equilibria in the game must involve some chance of each side getting punished. This means the third party must be indifferent between punishing each side in equilibrium. The payoff for punishing player 1 must be equal to the payoff for punishing player 2, so g1 – g2 = g2– g1 + b, so g1 = g2 + b/2. In terms of the likelihoods of committing the crimes, this condition is p1/(p1+p2) = p2/(p1+p2) + b/2, or p1 = p2 + b/2*(p1+p2), or (1-b/2) p1 = (1+b/2)p2, or p1 = p2(1+b/2)/(1-b/2). Since the term containing the bias parameter is greater than 1, we know that in equilibrium p1 > p2. This means that player 1 is more likely to commit the crime than player 2, the side the third party is biased against. Therefore, bias in a third party encourages the side it is biased in favor of to commit provocations.
Finally, Each of the players must also be indifferent between committing the crime and not. Let the likelihood that the third party imposes sanctions on player 1 be s1 and for player 2 s2. For player 1, the payoff for committing the crime is –s1 + s2, which must equal the payoff for not committing the crime, which is zero. So we know that s1 = s2, and the two sides must be equally likely to be punished in equilibrium. The same analysis holds for player 2.