Political Game Theory

From The Chapterhouse Codex

Game theory is the mathematical analysis of strategic interaction. In the sixty years since the appearance of von Neumann and Morgenstern's classic Theory of Games and Economic Behavior (Princeton, 1944), game theory has been widely applied to problems in economics. Until recently, however, its usefulness in political science has been underappreciated, in part because of the technical difficulty of the methods developed by economists.

1 Notes
- 1.1 Election Deadlock
- 1.2 Strategic Deterence
2 Refrences

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Notes

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Election Deadlock

Let Jones and Smith be the only two contestants in an election that will end in a deadlock when all votes for Jones (J) and Smith (S) are counted. What is the expectation value of:

$X_k \equiv |S-J|$

after $k$ votes are counted? The solution for any natural number $k$ is:

$\begin{matrix} \langle X_k \rangle= \frac{k(2N-k+k \bmod 2)}{2N} {N \choose (k-k \bmod 2) / 2}^2 {2N \choose (k-k \bmod 2)}^{-1} \end{matrix}$

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Strategic Deterence

Models of deterence must take into account that the probability that the threat will be carried out is such that the utility of the challenger backing down is maximised and the utility of the threat being carried out is minimized. Otherwise stated, a threat must be realistic and devistating while a rewarding way out is presented for the challenger.

$p > \frac{u_{CH}(CS) - u_{CH}(BD)}{u_{CH}(CS) - u_{CH}(TC)}$

Where $u C H (C S)$ is the utility of the challengers success (threat will not be carried out), $u C H (B D)$ is the utility of the challenger backing down, $u C H (T C)$ is the utility of the threat being carried out (challenger failed), and p is the credibility of the defender's intention to carry out the threat. The challenger will press ahead unless p is greater than the critical risk to the challenger (right hand side of the inequality).

As $p$ decreases $u C H (T C)$ increases. The inverse is also true, as $p$ increases $u C H (T C)$ decreases. The relationship is a smooth function represented as the product of the constant I (impact of the threat) and $1 - p$ :

$u_{CH}(TC) = I(1 - p)\,$

While it s tempting in deterence models to simplify $u C H (C S)$ and $u C H (T C)$ using von Newmann Morgenstern to 1 and 0 respectively, it is a misconception to not believe that it is possible to have $(T C) P (B D)$ . This is espically true when $B D$ involves a new restriction, such as economic sanctions, or when $I$ approaches $0$ , such as a threat that can be easily defended against.

Additional restrictions involved in $B D$ , such as retaliation, can be expressed by inserting a probability $q$ , representing the chance of retaliation or sanctions, into the model, such that:

$u_{CH}(BD) = (1-q)\widehat{BD}$