Victor V. Motti Articles

The Prospect of USA and Iran conflict: Insights from game theory

After the withdrawal of Trump administration from the Iran nuclear deal the tension has been rising in the Middle East and in particular the Persian Gulf and we are approaching the armed conflict stage. We can apply game theory to explore the possible equilibrium points between the US and Iran. We need to consider a range of current strategies on both sides, the payoffs of each player or opponent, and determine the strategic choices for them.

In this framework we assume that there are two key players namely the USA and Iran.

Available strategies for the USA are: Negotiation, Sanction, and War.

And

Available strategies for Iran are: Negotiation, Defiance, and War.

As one can observe in the table below by crossing these strategies nine possible scenarios can be identified.

 Iran USA Negotiation Defiance War Negotiation Negotiation Negotiation Negotiation Defiance War Sanction Sanction Sanction Negotiation Defiance War War War War

In order to quantify this game space, we need to assign payoffs, or utilities, to each of these nine scenarios from the vantage point of each of the players.

For instance, for Iran, the scenario of (defiance, sanction) is preferred to the scenario of (negotiation, negotiation). Also, the scenario of (defiance, negotiation) is preferred to the scenario of (negotiation, sanction).

On the other side, for the USA, the scenario of (negotiation, negotiation) is preferred to the scenario of (defiance, sanction). Also, the scenario of (defiance, war) is almost the same as or indifferent to the scenario of (defiance, sanction).

After examining all the scenarios pairwise to establish the preference pattern per each player, which in total span 36 pairs, and supposing that the range of the payoff is from -2 to +2, we can prepare the table below in which the payoffs have been assigned to all scenarios for each side of the conflict.

 Iran Negotiation Defiance War USA Negotiation -1 2 -1 2 -1 0 Sanction -1 1 0 1 1 0 War -2 0 1 2 1 2

A quick assessment reveals that this game has two equilibrium scenarios. In other words, scenarios that are locked and neither side of the conflict, or the players, are urged to change their strategy unilaterally to increase their payoff. The equilibrium scenarios are: (defiance, sanction) and (war, war).

Therefore, we can delete both the row and the column of negotiation from the above matrix. In the next step after the negotiation strategy is removed for each player because equilibrium points are not there, we can look at the simplified matrix below.

 Iran Defiance War USA Sanction 1 0 1 0 War 0 1 1 2

In the real world, we seldom have pure strategies, and instead look at mixed strategies per each player. In other words, there is uncertainty involving the choice of each strategy by the opponent in the game and we need to consider probabilistic combinations if possible. Hence, each player should calculate the payoff of each own strategy based on a probability distribution over the strategies of the opponent.

Let’s consider the first case. Iran should make a conjecture about the US playing sanction or war and then on this basis needs to compare the payoffs of the defiance and war. Suppose that the probability of sanction by the US against Iran is P. Then the probability of war by the US is (1-P). Now we can calculate the payoffs of defiance and war separately as shown below.

Defiance Payoff =(1×P) + (0×(1-P)) = P

War Payoff = (0×P) + (1×(1-P)) = 1-P

The question is under what circumstances these payoffs are equal or indifferent. Through this we can figure out the threshold probability for the flip of preference.

Defiance Payoff = War Payoff

P = 1-P; P = 0.5

This calculation indicates that if the probability of sanction by the US is 70% and the probability of war is 30%, because the threshold is 50%, for Iran then defiance is preferred over war. But if the probability of war by the USA is more than 50% then for Iran war is preferred over defiance. In other words, this quantitatively validates our qualitative logical and intuitive current sense making of the conflict prospect.

However, the second case is more illuminating. The USA should make a conjecture about Iran playing defiance or war and then on this basis needs to compare the payoffs of the sanction or war. Suppose that the probability of defiance by Iran against the US is Q. Then the probability of war by Iran is (1-Q). Now we can calculate the payoffs of sanction and war for the US separately as shown below.

Sanction Payoff =(1×Q) + (0×(1-Q)) = Q

War Payoff = (1×Q) + (2×(1-Q)) = 2-Q

The question again is under what circumstances these payoffs are equal or indifferent because that will allow us to find the threshold probability for the flip of preference.

Sanction Payoff = War Payoff

Q = 2-Q; Q = 1

This calculation indicates that if the probability of defiance by Iran is 100% and the probability of war is 0%, because the threshold is 100%, for the US, sanction is indifferent to war.

The above calculation uncovers two key insights. First, even if Iran focuses on defiance then the US has not a significant preference for more sanctions instead of war. And second, and a deeper insight about the prospect of conflict, is that if the US believes that there is an insignificant probability of war by Iran, say 1%, then the payoff of war strategy is more than that of sanction. In brief, if our assumptions and simplification used in the theoretical framework are true and defendable, then we should anticipate that a minor indication by Iran of a will to war escalates and the US will also enter the war phase.