Applied Mathematics can help up visualize complex patterns by reducing them to core elements.

So in a hypothetical insurgency, there are 5 categories of players with their order of preferences. This models two choices – Negotiate or Fight. Each has a “win-lose” utility value.

Insurgents:
True Believer – War, Negotiations, Repression
Reluctant Soldier – Negotiations, War, Repression
Complacent Opponent – Negotiations, Repression, War
Government:
Responsive Government – Negotiations, Repression, Surrender
Repressive Government – Repression, Negotiations, Surrender

In this basic game, a responsive government may be able to negotiate with Reluctant insurgents and complacent opponents. True Believers are not favorable to negotiation, but the threat of defeat may be so severe that they choose not to go to war.

What are they negotiating for? We have to attach a utility value to their demands. Some negotiations may be coordination games. Both sides agree to a compromise where they can both equally benefit and stop the violence. Insurgents may have reasonable demands – they reject high levels of taxes or exclusion from trade. Governments can negotiate with them to end the threat of violence.

Let’s say the civil violence starts over a trade dispute. This is a coordination game.
Exclusion from trade: Insurgents (0), Government (3)
Included in trade: Insurgents (3), Government (3)
War (Insurgent win): Insurgents (2), Government (-1)
War (Government win): Insurgents (0), Government (2)

Negotiation has higher utility than war in this game. There is a coincidence of interest if both sides agree to trade. The Government gains nothing by excluding opponents from trade, but the government faces losses of trade if insurgents rebel. Insurgents have reasonable demands.

In other games, negotiation is asymmetrical and disadvantageous to one side. This are distributional problems and zero-sum games. There is no easy compromise.

Let’s say the Insurgency wants an Islamist Government with the implementation of Sharia Law. The Secular Government wishes to retain secular laws. This is a zero-sum game.
Under Sharia Law: Islamists (+3), Government (-3)
Under Secular Law, Islamists (-3), Government (+3)
At War, each side receives 0 utility.

There is an incentive to reject unfavorable laws to go to war. The Insurgent demands are unreasonable even for supporters of a Responsive Government. The Governments receives negative utility by negotiating with Insurgents but a high positive utility for defeating the insurgents. This changes their order of preference to favor warfare and repression of Islamists.

The game grows more complex when we model individual game decisions. Let’s say there are 200 individuals – 100 in the government forces and 100 insurgents. They are not evenly distributed. Insurgents forces are 5% True Believers, 15% Reluctant Soldiers, and 80% Complacent Opponents. Complacent Opponents are not actively fighting, but they may be giving morale and material support to the insurgents if they are repressed. Government forces are 60% Responsive and 40% repressive.

This changes the mechanics of the game. A dynamic game occurs over many turns where players change strategies. Individuals may change allegiances from one category to another. A repressive government may push complacent opponents to become reluctant soldiers. A responsive government may encourage complacent opponents to join the responsive government. True Believers may attack a Responsive Government and turn it into a Repressive Government.

One last factor to model: Probability. Each side is uncertain what the results of war or negotiations will be. Every decision is a competition between asymmetrical powers with a high level of risk and uncertainty. Insurgents may choose war over negotiations if they believe they have a high probablity of winnign the war. They may be incorrect and in hindsight realize that early negotiations would have had the highest payoff.

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