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title: War as Bargaining
subtitle: POSC 3610 -- International Conflicct
author: Steven V. Miller
institute: Department of Political Science
titlegraphic: /Dropbox/teaching/clemson-academic.png
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# Introduction
### Goal for Today
*Introduce students to thinking rationally and strategically in world politics.*
# Strategy and Extensive Form Games
### International Interaction Game
![International Interaction Game (via BdM and Lalman, 1992)](iig.png)
### Possible Outcomes of the Game
1. The status quo
2. A negotiated settlement
3. A (or B) acquiesces.
- i.e. one side concedes the issue without being attacked.
4. A (or B) capitulates.
- i.e. one side concedes the issue *after* a preliminary attack.
5. A (or B) retaliates to an attack.
- i.e. both sides fight a war.
### Assumptions of the Game
1. Decision-makers are rational and strategic (recall previous lecture).
2. *p*=1 or *p*=0 **only** for acquiescence, capitulation, or status quo.
- i.e. the utility of all other outcomes is weighted by probability.
3. The utility of negotiation or war is a lottery
- $p_A$, $p_B$ = probability of "winning" the lottery
- $1 - p_A$, $1 - p_B$ = probability of "losing" the lottery.
- Do note these are not identical variables.
4. Each state leader prefers negotiation over war.
- This is also common knowledge.
### Assumptions of the Game
5. Violence involves costs *not* associated with negotiations.
- Capitulation: the capitulating state eats the costs of the attack.
- This also implies a first-strike advantage.
- Any attack: the attacking state incurs costs associated with failed diplomacy.
6. Both A and B prefer any policy change to the status quo, but: $SQ_i > ACQ_i$.
7. Foreign policies follow domestic political considerations.
- These may or may not include consideration of international constraints.
### Additional Restrictions of the Game
These assumptions imply the following preference restrictions.
- SQ > Acquiescence or capitulation by A (or B).
- Acquiescence (by the opponent) is most preferred outcome.
- Acquiescence by $i$ > Capitulation by $i$.
- Negotiation > Acquiescence/capitulation/an initiated war by $i$.
- Capitulation by $i$ > Initiated war from $j$
- War started by $i$ > War started by $j$
- Capitulation by $i$ > zero in negotiations
- War started by $j$ > zero in negotiations.
### Interesting Implications of IIG
War is the complete and perfect information equilibrium *iff* (sic):
1. A prefers to initiate war > acquiescence to B's demands.
2. A prefer to capitulate, but B has a first-strike advantage.
3. B prefers to fight a war started by A rather than acquiesce to A's demands.
4. B prefers to force A to capitulate rather than negotiate.
- We call this a "hawk" in this game.
- A "dove" prefers negotiations over a first-strike.
### Interesting Implications of IIG
*Uncertainty doesn't automatically lead to higher probability of war.*
- If A mistakes that B is a dove (when, in fact, B is a hawk) and
- B mistakenly believes A would retaliate, if attacked. Then:
- A offers negotiation to B.
- B responds with negotiation to A.
### War as Failed Bargain
However, even the IIG misses that wars are failed bargains
- States have numerous issues among them they try to resolve.
- They may use threats of force to influence bargaining.
- If bargaining fails, states, per our conceptual thinking, resort to war.
*However, there is conceptually a range of possible negotiated settlements both sides would prefer to war.*
### A Simple Model of Crisis Bargaining
To that end, we devise a simple theoretical model of crisis bargaining.
- There are two players (A and B).
- A makes an offer (0 $< x <$ 1) that B accepts or rejects.
- If B accepts, A gets $1 - x$ and B gets $x$.
- If B rejects, A and B fight a war.
### A Simple Model of Crisis Bargaining
The war's outcome is determined by Nature (*N*)
- In game theory, Nature is a preference-less robotic actor that assigns outcomes based on probability.
- If (A or B) wins, (A or B) gets all the good in question minus the cost of fighting a war ($1 - k$)
- Assume: $k > 0$
- Costs could obviously be asymmetrical (e.g. $k_A$, $k_B$), but it won't change much about this illustration.
- The loser gets none of the good and eats the war cost too ($-k$).
We assume minimal offers that equal the utility of war induce a pre-war bargain.
### A Simple Model of Crisis Bargaining
Here's a simple visual representation of what we're talking about.
### Solving This Game
How do we solve this game? How do A and B avoid a war they do not want to fight?
- The way to solve extensive form (i.e. "tree") games like this is **backwards induction**.
- Players play games ex ante (calculating payoffs from the beginning)
rather than ex post (i.e. hindsight).
- They must anticipate what their choices to begin games might do as
the game unfolds.
In short, we can solve a game by starting at the end and working
back to the beginning.
### Solving This Game
For our purpose, we need to get rid of Nature.
- Nature doesn't have preferences and doesn't "move." It just assigns outcomes.
- Here, it simulates what would happen if B rejected A's demand.
We can calculate what would happen if Nature moved by calculating the expected utility of war for A and B.
### Expected Utility for A of the War
EU(\textrm{A} | \textrm{B Rejects Demand)} &=& (1 - p)(1 - k) + p(-k) \nonumber \\
&=& 1 - k - p + pk - pk \nonumber \\
&=& 1 - p - k \nonumber
In plain English: A's expected utility for the war is the probability (1 - p) of winning the war, weighting the value of the good (i.e. 1), minus the cost of war (k).
### Expected Utility for B of the War
EU(\textrm{B} | \textrm{B Rejects Demand)} &=& (1 - p)(-k) + p(1 - k) \nonumber \\
&=& -k + pk + p - pk \nonumber \\
&=& p - k \nonumber
In plain English: B's expected utility for the war is the probability (p) of winning the war, weighting the value of the good (i.e. 1), minus the cost of war (k).
### The Game Tree, with Nature Removed
### Solving This Game
Now, continuing the backward induction, we focus on B.
- B ends the game with the decision to accept or reject.
- B does not need to look ahead, per se. It's now evaluating whether it maximizes its utility by accepting or rejecting a deal.
### Solving This Game
Formally, B rejects when $p - k > x$.
- It accepts when $x \ge p - k$.
- Notice A has a "first-mover advantage" in this game.
- This allows it to offer the bare minimum to induce B to accept.
- It would not offer anymore than necessary because that drives down A's utility.
We say A's offer of $x = p - k$ is a minimal one for B to accept.
### Solving This Game
Would A actually offer that, though?
- In other words, are $x = p - k$ and $1 - x \ge 1 - p - k$ both true?
Recall: we just demonstrated $x = p - k$. From that, we can say $1 - x \ge 1 - p - k$ by definition.
- The costs of war ($k$) are positive values to subtract from the utility of fighting a war.
### The Proof
What A would get (1 - *x*) must at least equal 1 - *k - p*. Therefore:
1 - x &\ge& 1 - k - p \nonumber \\
1 - 1 + k + p &\ge& x \nonumber \\
p + k &\ge& x \nonumber
### Solving This Game
We have just identified an equilibrium where two states agree to a pre-war solution over a contentious issue.
- There exists a bargaining space where A and B resolve their differences and avoid war.
![The Bargaining Space](bargaining-space.png)
# Conclusion
### Conclusion
War is a form of bargaining failure. It never happens in a world of complete/perfecct information, except for:
- Issue indivisibility
- Incomplete information
- Commitment problems