Todd Sandler on Game Theory, Terrorists, and International Relations 2/3
So you’ve been reading up on Game Theory on this blog and have decided to keep going on. In this post, I continue with Todd Sandler’s paper, specifically on the coordination dilemma. Before reading this paper, I was not familiar with these theories, I hope to break down Sandler’s thoughts to create a better understanding of these dilemmas.
Fist up is the Coordination Dilemma, specifically the freezing of terrorists’ assets
“A common game form for some policy choices is a “stag-hunt” assurance game, where both parties are best off if they take identical measures. When a player takes the measure alone, this player receives the smallest payoff and the player who does nothing earns the second-greatest payoff. This kind of scenario is descriptive of a host of counterterrorism policies where two or more nations must act in unison for the best payoffs to result. Examples include freezing terrorist assets, denying safe haven to terrorists, applying sanctions to state-sponsors, or holding to a no-negotiation policy. Even one nation that breaks ranks can ruin the policy’s effectiveness for all others. To illustrate such scenarios, I use freezing terrorist assets as a generic example and begin with a two-nation symmetric case.”
Remember this post, where I talk about ’stag hunts.’ Imagine two hunters deciding to take down a stag or chase rabbits. Both hunters are needed to kill the stag, but only one to kill a rabbit. If one hunter hunts the stag while the other chases the rabbit, the chaser gets the highest payoff and the hunter gains nothing. The author will now delve down into the grid that is below.
“Matrix a in the top of Figure 3 displays this scenario where the highest payoff of F results from mutual action, followed by a payoff of A from doing nothing either alone or together. The smallest payoff, comes from freezing assets alone since the terrorists can merely transfer their assets elsewhere, leaving the acting country with some banking losses but few safety gains. Since A > B, there is no dominant strategy. There are, however, two pure strategy Nash Equilibriums: both countries freeze assets or neither freezes assets q of nation 2 freezing terrorist assets that make nation 1 indifferent between freezing terrorist assets and doing nothing. Similarly, I derive the probability p of action on the part of nation 1 that makes nation 2 indifferent between the two strategies. Once p and q are identified, equilibrium probabilities for maintaining the status quo simply equal 1 − q and 1 − p for nations 2 and 1, respectively. The relevant probabilities are indicated for matrix a besides the respective column and row.”
Remember, the highest payoff is F, but A > B. So countries should freeze assets. However, if nation 1 knows that nation 2 does nothing, the only way nation 1 gets a positive payoff is by also doing nothing. The key for nations is to get F or A, but never B.
Figure 3. Alternative freezing assets scenarios.
“A third Nash equilibrium involves mixed strategies in which each pure strategy is played in a probabilistic fashion. To identify this mixed-strategy equilibrium, I determine the probability q of nation 2 freezing terrorist assets that make nation 1 indifferent between freezing terrorist assets and doing nothing. Similarly, I derive the probability p of action on the part of nation 1 that makes nation 2 indifferent between the two strategies. Once p and q are identified, equilibrium probabilities for maintaining the status quo simply equal 1 − q and 1 − p for nations 2 and 1, respectively. The relevant probabilities are indicated for matrix a besides the respective column and row. The calculations for q (or p not shown) go as follows:
qF + (1 − q)B = qA + (1 − q)A, (1)
from which we have
q = (A − B)/(F − B). (2)when q exceeds this value, cooperation in the form of both countries freezing terrorist assets is the best strategic choice. An identical expression holds for p owing to symmetry. The ratio in (2) represents the adherence probability that each nation requires of the other to want to coordinate its freeze policy. A smaller equilibrium probability favors successful coordination, because a nation requires less certainty of its counterpart’s intention to freeze assets in order to reciprocate.”
If q becomes a smaller ratio, two countries would want to cooperate and freeze terrorist assets. The equation states that if q = (Status Quo minus Freezing assets alone) divided by (Freezing assets together minus Freezing assets alone) approaches (1), countries are most likely to participate. This happens because countries want to drop the ratio below (1), so they want to increase the number on the bottom of the equation. In this case, Freezing assets together (F).
From Equation (2), either a larger gain (F) from a mutual freeze or a smaller status-quo payoff (A) promotes the coordination equilibrium by reducing the required adherence probability. 9/11 raised F and lowered A because nations realized the benefits from limiting terrorists’ resources and the catastrophic consequences that inaction could have for everyone. In the last few decades, the increase of carnage from terrorists attacks increasingly draws nations to coordinate counterterrorism activities when needed. 9/11 was quite catastrophic and it led to many nations participating in freezing assets, but this participation is by no means universal. Differentiating the right-hand side of (2) with respect to B shows that a decrease in the payoff associated with unilaterally freezing assets inhibits cooperation by raising p or q.
Big attacks (9/11, 7/7, Madrid) help nations cooperate with each other because they have both already lost so much because of the big attack. These sobering experiences put the attacked nation at a disadvantage and therefore has to cooperate to catch up.
This game scenario can be readily generalized to n homogeneous nations, where at least n nations must freeze assets if each participant is to receive a payoff of F. For less than n freeze participants, each adherent receives B for cooperating and the nonadherents get A. If nations are uncertain about the intentions of other nations, then freezing assets is a desired policy provided that a nation believes that the n − 1 required additional participants will follow through with a collective probability greater than q. This then implies that each nation must be expected to cooperate by at least the n − 1st root of q, which for even modest groups may require near certainty. This is not an encouraging finding. If, however, the required number of adherents for coordination gains can be limited, then this decreases the assurance probabilities. For an agreement to freeze assets, this is best accomplished by first unifying some of the major financial-center nations – i.e., the United States, the United Kingdom, Switzerland, Japan and Germany. A concern with this strategy is that some near-catastrophic terrorist acts are not very costly – e.g., the 1993 WTC bomb cost just $400 and caused $500 million in damages (Hoffman, 1998) – so that near-universal freezes may be required.
Countries will only cooperate if they feel that a set amount of countries will also participate. A country doesn’t want to pioneer freezing terrorist assets and allow a bunch of other countries to simply ride on their coattails. However, this way of thinking is not conducive to innovation or success.
In matrix b in Figure 3, an alternative scenario is displayed where not freezing assets, when the other nation freezes, gives the second highest payoff to the noncooperator, so that F > E > A > B. This scenario implies that the nation that does not join the freeze can profit by providing a safe haven for terrorists’ funds. The nation may be motivated to do so if it does not view its own people or property as likely targets of the terrorists. The two pure-strategy Nash equilibriums are for a mutual freeze or no action along the diagonal of the matrix. For the mixed-strategy Nash equilibrium, the adherence probabilities are now:
p = q = (A − B)/[F − B + (A − E)], (3)
which are greater than those in (2), because (A−E) < 0. Hence, coordinating a freeze becomes more difficult owing to profitable opportunities available to less scrupulous nations that can greatly limit gains from action to freeze assets, eliminate safe havens, or abide by no-negotiation pledges (see, e.g., Lee, 1988). Policies that penalize noncompliance can reverse the ranking of A and E, so that A > E. As a consequence, adherence becomes easier to achieve. There are two practical problems: (i) to identify nations that accept terrorists’ funds and (ii) to convince nations to punish nonadherents. Since nations hide their bad behavior, singling out actions for punishment is not so easy. Imposing sanctions is itself a prisoners’ dilemma game that presents its own collective action concern.
It is never a good thing for a country to benefit most from providing a safe haven to terrorists. In this case, more powerful countries need to step in (with sanctions, military threats, etc.) to prevent the situation from becoming worse.
In Figure 4, a final freeze scenario allows for asymmetry where nation 2 has more potential nonadherence profits but fewer gains from acting alone than its counterpart. That is, I assume that F > Ei > A > Bi , i = 1, 2, where E2 > E1 and B1 > B2. The pure-strategy Nash equilibriums are still the matching-behavior outcomes along the diagonal of the matrix in Figure 4. For the mixed-strategy equilibrium, the adherence probabilities are:
To act, nation 2 needs greater assurance than nation 1 that the other nation will freeze assets. Such asymmetry is likely to work against consummating a freeze.
When coordination is required for a counterterrorism measure, many factors work against getting sufficient action. A crucial consideration is the minimum number of nations required for coordinating antiterrorism activities. As this minimum increases, nations must have greater assurance that others will cooperate for them to follow suit. Any policy action that limits this minimum bolsters cooperation. As the threat of terrorism escalates, coordination of counterterrorism is encouraged because cooperative outcomes have greater payoffs and unilateral action has smaller payoffs. The application of technology to track money flows can identify duplicitous nations that hamper other nations’ actions by providing safe havens to terrorists’ assets. Retribution against these “spoiler” nations can foster more fruitful coordination by sending a clear signal that profiting from terrorism has consequences. Efforts by the International Monetary Fund and World Bank to assist countries in tracking asset transfers can lower the costs of unilateral action, thereby boosting efforts to freeze terrorists’ assets.
Figure 4. Asymmetric freezing assets scenario.
The final point made in this post is that the number of countries participating in the freezing a terrorist assets should be small. This way, countries get a higher payoff and are more likely to partake. As n (the number of nations participating) increases, the payoff decreases and countries are more likely to not participate. The author says that regulations or policies can aid in that effort.
So, now we are getting down into the dirty parts of Game Theory. There is one more post to come on this topic, about the Public Choice Dilemma. Some of this stuff can get a little confusion, especially the bits and pieces of the equations, but reading over it a few times helps for the information to sink in. See you soon!
