Bargaining Model of War

Bargaining Model of War:     

Now, lets begin with some (strong) assumptions- centered on the page and numbered:

1. Two states A and B

2. Dispute over an object worth that is standardized to 1

The 1 equals 100% over the object. Where the object can be anything disputed over by both states, whether it be land, resources, or infrastructure.

i.e. An oil field valued to be $80 million equals 1

The Amazon rainforest in its entirety is equal to 1

3. Power remains stable through time

No external forces can act on the probability of victory

4. Everyone knows each other's probabilities/payoffs

5. Object is infinitely divisible 

6. No splendid first strike advantages 

First strike is the ability to destroy most of the retaliatory response and achieve minimal damage to self.

P(A) is the probability where state A wins the war

P(B) is the probability where state B wins the war

          

7. There are no draws, so P(A) + P(B) = 1

P(A) and P(B) are expectations.

8.If the states fight a war they will both incur costs that reflect absolute costs and resolve.

Absolute costs constitute souls lost, property damage, and wartime economic disruption

Resolve constitutes the value of how much a state cares about the issue (interest) and will decrease the cost.

A state with a higher resolve over an issue is willing to incur higher absolute cost over that issue.

For instance: Scotland has a higher resolve over Edinburgh than China does and would be willing to put up more absolute costs for proprietorship. (because China does not care about Edinburgh) 

         

Let C(A) be the expected cost that state A incurs over the war divided by the total amount of the object and standardized. 

Let C(B) be the expected cost that state B incurs over the war divided by the total amount of the object and standardized. 

          

       9. C(A)>0 and C(B)>0, the cost of war will never outweigh the value of the war

Let E(A) and E(B) denote the expected outcomes of war for State A and State B respectively in a winner-take-all assumption. God bless Thomas Bayes. 

E(A) = (P(A))*1 + (1-P(A))(0) -C(A)

         = P(A)-C(A)

E(B) = (P(B))*1 + (1-P(B))(0) +C(B)

         = P(B)+C(B)

C(B) is added to P(B), rather than subtracted, because if we look at it from B's perspective a positive cost will decrease capital. 

Basically, the expected outcome from war is the probability of winning the war minus the cost of fighting.

Seemingly, a state that has a higher expected probability of winning a war will incur lesser costs in expectation. For generality (using basic algebra), on answering the next question, let us not create a function for this.

Provided with these assumptions, is it possible to have a negotiated contract that is a viable alternative to war for both states simultaneously? 

Answer: Yes, but only if a negotiated contract yields a greater outcome than the expected value of conducting war for both states. 

This leads to the following peace constraints, but lets look at the costs first.

State A's Peace Constraint

   

Let x be State A's share of the bargained settlement, which has been standardized to 1, the complete value of what is being waged on, so 0<=x<=1

State A is content if x>= E(A)>=P(A)(1)-C(A)

                                 x>=E(A)>=P(A)-C(A)

In words, State A is satisfied if the probability of winning a war minus the cost is less than the negotiated contract x. Satisfaction implies bargain, i.e. not fighting. Makes sense, doesn't it?

State B's Peace Constraint

1-x is State B's share of bargained settlement, because State B gets everything that State A does not have

State B is content if 1-x>=1-E(B)>=P(B)(1)-C(B) 

                                 1-x>=1-E(B)>=P(B)-C(B)

                                  x<= E(B)<=1-P(B) + C(B)

In words, State B would rather fight than allow State A to take more than the cost that State B would incur. 

Combining these we realize that State A and State B can reach a mutually satisfactory agreement if there exists an x, such that x exists between the cost of war for State A and the cost of war for State B.

P(A)-C(A)<=x<=1-P(B)+C(B)

From the earlier assumption we know, P(A)+P(B) =1

which implies P(B)=1-P(A)

P(A)-C(A)<=X<=1-(1-P(A))+C(B)

P(A)-C(A)<=P(A)+C(B)

-C(A)<=C(B)

C(A)+C(B)>=0

C(A)+C(B)>0 

(the greater than or equal sign drops from earlier assumptions because C(A)>0 and C(B)>0)

Ok... That is great, but how do we interpret this result?

Basically(precisely), a bargained settlement can exists if the cost of war for either State A or State B is greater than zero.

And the bargaining range is mutually preferable to war.

Thinking back to the definition of x: x represents the total amount amount of an object standardized between and knowing that it exists between 0 and 1, it is possible to visualize a peace interval.. As follows:

Now lets add one more assumption to our model. 

source: http://giphy.com/search/dr-strangelove

-Suppose the same as before, but now Suppose State A controls the entire object at the start and has to negotiate with what State B wants on a take-it-or-leave-it offer.

-If State B accepts, the settlement is contracted. Otherwise, the states fight. 

This gives us a "Crisis Bargaining Model"

This is just a way to model the algebra that we just did. 

If State B accepts the contract, State A earns x and State B gets 1-x. State A payoffs are in red, while State B payoffs are in blue. 

If State B rejects the contract we need to calculate State A's  expected payoff and State B's expected payoff through backwards induction. We have calculated these numbers previously and I will do the backwards induction in class. 

E(A)=P(A)-C(A)

E(B)=1- P(A)-C(B)

State B is willing to accept the contract if the payout from accepting is larger than the payout from rejecting. The converse is true. Let's see what State B does by setting up an inequality.

1-x>= 1-P(A)-C(B)

-x>=-P(A)-C(B)

x<=P(A)+C(B)

Thus, as long as the demand from State A is less than the cost of war than State B will accept. 

Let's look at it from State A's Perspective, there are two options. Remember that State A's payoff is increasing in x so the larger the x the better. So instead of offering an x<=P(A) +C(B), State A will offer an x=P(A)+C(B), which is the maximum amount of x that State A can get without State B rejecting. Anything larger, State B will reject. 

So,

What is better for State A?

1. x=P(A) + C(B) ----> State B accepts offer----> State A earns x

2. x> P(A) + C(B)----> State B rejects offer----> State A earns war payoff= P(A)-C(A)

P(A)+C(B)>P(A)-C(A)... So option 1 is better! Look at the 0 to 1 scale.

State A is able to gain the loss that State B would incur during a war through the settlement. 

Relaxing these assumptions gives us a more rationalist explanation for most wars because, afterall, this model only shows peaceful solutions and does not explain other phenomenon that have happened throughout history. However, this model can be nicely applied to the Cold War.

Question: Why was the Cold War never a hot war?

The most simplistic answer that I would give is: Nuclear Weapons. 

"How I stopped worrying and learned to love the bomb"- Dr. Strangelove

source:http://giphy.com/gifs/stanley-kubrick-peter-sellers-dr-strangelove-or-how-i-learned-to-stop-worrying-and-love-6zE5U2oxKzCw

But before getting into that I would like to give a brief history lesson. At the end of World War II Europe was split along a line known as the Iron Curtain, separating the "Western World" and the USSR. 

source: http://kirycoldwar.weebly.com/uploads/2/9/9/5/29953109/877879467.jpg 

Two prominent powers emerged... The United States and The Soviet Union. At first the United States was the only country to have the power of nuclear weapons, but an arms race began and the Soviet Union soon gained nuclear weapons after. 

The power of nuclear weapons is truly unfathomable and the stockpiles of each super-power had the capacity to destroy the world 100-fold. 

The presence of two super-powers brought to life a theory known as Mutually Assured Destruction (MAD).

Mutually Assured Destruction occurs if:

1. Both states are self-preserving (States aren't actively trying to be vanquished)

2. Both states have stockpiles of nuclear weapons

3. Each state has a secured second strike; no state can achieve a splendid first strike 

Second strike: having the capability to launch a nuclear retaliatory strike after being struck.

Splendid first strike: being able to destroy most of the retaliatory response and achieve minimal damage to self.

Under these assumptions, no side would want to start a large-scale war because no side would win.

If the United States started a war, the United States would face enormous nuclear retaliation from the Soviet Union. (Second Strike)

Likewise, if the Soviet Union started a war, the Soviet Union would face enormous nuclear retaliation from the United States. (Second Strike)

With that information it should be understood that the cost of war increases dramatically. An increased cost of war dramatically increases the bargaining range of a mutual agreement, thus it is much more likely that a mutual agreement occurs.

Bargaining range is x in:

P(A)-C(A)<x<P(B)+C(B)

Another way to view this information is through a Nash Equilibrium.

It can be shown through payoff in a Nash Equilibrium:

(USA,USSR)	Peace	Nukes
Peace	(1,1)	(0,4)
Nukes	(4,0)	(0,0)

Above is a Nash Equilibrium table and it shows the payoffs for the following scenarios. I believe Greg will be doing Nash Equilibriums in Submarine vs. Destroyer, as well as myself solely on the topic in another blog so I will not be getting into detail on the blog (but I will in class!). This one specifically implies that there is no winner to a nuclear war between countries who both are able to nuke.

Nuclear war means death for just about everyone as we know it.

Due to this stalemate of nuclear weapons, the United States needed to resort to other uses of power to deter the Soviet Union from encroaching into Western Europe.

A policy of the United States was extended deterrence, which is the use of weaponry or presence to deter attacks on allies. The United States was allied with any participant in NATO and, for that reason, had interest in Western Europe.

The United States never official used Mutually Assured Destruction as a policy, but it did use the threats of conventional and targeted nuclear strikes to deter the Soviet Union from invading Western Europe as we know it.

As Dr. Jockel said, "The best way to play chicken is to put on a blindfold and hold whiskey bottles."

Chicken is a game with two participants who drive cars at each other and the first one to swerve out of the way loses.

This quote from Arms and Influence is similar, "Manipulating the shared risk of war. It means exploiting the danger that somebody may inadvertently go over the brink dragging the other with him. If two climbers are tied together, and one wants to intimidate the other by seeming about to fall over the edge, there has to be some uncertainty or anticipated irrationality over it" source 2

Basically you pretend to be crazy and your opponent has no choice but to believe your bluff.

The United States was able to deter the Soviet Union from entering Western Europe through the bluff of chicken.

Now its time for some food for thought...

Would the proliferation of nuclear weapons stabilize the world and bring about peace/mutual agreements instead of war?

-How should the United States react to a Russian invasion of the Baltic states?

-How does Mutually Assured Destruction jeopardize extended deterrence?

Bibliography:

1.Spaniel, William. Game theory 101: the rationality of war.

Chapter 2: War’s Inefficiency Puzzle

2. University, Harvard, and Thomas C. Schelling. Arms and influence. Greenwood Press, 1976.

page 99

3.JimBobJenkins. "International Relations 101 (#19): Crisis Bargaining." YouTube. YouTube, 20 Aug. 2012. Web. 13 Mar. 2017.