Wartime Elections

Motivation

How does war (or any highly significant policy) affect the electorate’s vote choice? Moreover, how do different signals regarding the state of the war (success or failure) help the electorate to form their belief about the incumbent’s war policy? The political science literature regarding the affect of war on elections has a long history, beginning with John E. Mueller’s famous finding that as (the log of) casualties in war increases, favorable public opinion of the war decreases. Recently, Gelpi, Feaver, and Reifler have hypothesized that while the public is adverse to casualties, the tolerance for the human costs of war are shaped by two attitudes: (1) Whether the war justified and (2) Whether the war is likely to be a success – the later being more important than the former. See their book for the full argument.

We also know that the electorate often formulates their belief by using heuristics (cues) from elite sources. See Popkin’s The Reasoning Voter and Lupia and McCubbins’s The Democratic Dilemma (the latter uses a formal modeling and experimental approach)We also know from Zaller’s The Nature and Origins of Mass Opinions that public opinion is heavily influenced by elite discourse; however, for Zaller’s theory, elite cues are not necessarily differentiated (i.e. the effect of cues from the media, incumbent, and opposition party on the electorate is not investigated separately). My research is focused on separating multiple (and potentially competing) cues in the belief formation process.

Accurate vs. Biased Signals

I have developed a theory which looks at how different elite cues interact during the electorate’s belief formation about public policies which have especially high significance – with war being my primary focus. My signalling model blends objective but potentially inaccurate cues about the state of a policy (i.e. is the war on track for success or failure?) with biased cues from fully informed actors regarding the continuation of the policy (i.e. should the war be ended or not). The biased actor desires to get elected but also has preferences over the national interest . The noisy but objective signal comes from an actor I am calling the “Media”, while the biased signal comes form the “Opposition Party”.

Now I am sure that some people are already discounting my model because I am assuming that the “Media” is an unbiased actor. I don’t think this assumption is so crazy. I am mealy assuming that the electorate receives some objective (but possibly inaccurate) information from a news source. For example, casualties are reported, battle outcomes are reported, the taking of city is reported, etc. I would consider what some people call “New Media Sources” – which are often highly partisan and biased – to be a part of the actor I am calling the “Opposition Party”. These sources have a biased partisan opinion, but probably want what is best for the U.S.’s national interest as well. Thus, I think the assumptions of my model are reasonable. Even so, assumptions do not have to be perfectly accurate as long they are useful in telling us something interesting about the world. The empirical implications of my model will be judged by my experimental and large-N statistical tests (which I hope to post some info on soon).

The Model

The game begins with Nature (N) selecting the value of continuing the war, \omega . There are two states of the world, \omega = \overline{\omega} and \omega = \underline{\omega} , where \overline{\omega} is the value for a war that is worth continuing and \underline{\omega} is the value for a war which isn’t worth fighting. In the game, q is the probability that \omega = \overline{\omega} > 0, while (1-q) is the probability that \omega = \underline{\omega} < 0.

Next, N sends a noisy signal, s \in \{l,h\}, to the Media (M), about \omega. With probability \alpha \in (0.5,1), M receives an accurate signal and with probability (1-\alpha), M is given an inaccurate signal. Therefore, Pr(s = h | \omega = \overline{\omega}) = Pr (s = l | \omega = \underline{\omega}) = \alpha; in other words, \alpha signifies the probability that the state of the world matches the signal given by M. Also, Pr(s = l | \omega = \overline{\omega}) = Pr(s = h | \omega = \underline{\omega}) = (1-\alpha); thus, (1-\alpha) is the probability of  M being given an inaccurate signal which fails to match the state of the world. This means that, Pr(s = h) = \alpha q + (1- \alpha)(1-q), while Pr(s = l) = \alpha(1-q) + (1- \alpha)q. This implies that Pr(\omega = \overline{\omega} | s = h) and Pr(\omega = \overline{\omega} | s = l) follows from Bayes’ rule:

Pr(\omega = \overline{\omega} | s = h) = \frac{\alpha q}{\alpha q + (1 - \alpha) (1-q)}

and

Pr(\omega = \overline{\omega} | s = l) = \frac{(1 - \alpha)q}{(1 - \alpha) q + \alpha (1-q)}

Upon observing s, the Opposition (O), announces their desire to either continue or end the war, \rho \in \{c,e\}. I assume that \rho predicts what O would do if elected. O is electorally motivated but also values the national interest at a rate of \iota \sim u[0,\overline{\iota}].

In the final move of the game, the Electorate (E) decides whether to retain the incumbent or elect O. When E retains the incumbent, O receives \omega\iota while E receives \omega+\xi, where \xi \in [-1,1] represents the value of the incumbent relative to O with respect to economic or other domestic issues. This term may represent retrospective evaluations, prospective evaluations, differences in policy platforms, or combination thereof. The value of \xi is common knowledge.

If E elects O and O had set \rho=c, O receives \beta \in (0,1), which reflects the benefit of holding office plus \omega \iota , while E receives \omega. If O sets \rho=e, deciding to end the war, O receives \beta and E receives 0.

In the game, I assume that O knows \omega while E does not. However, E knows that O knows the true value of \omega, and may update their prior belief q to q_{s}^{\rho}, which reflects the information revealed by M and the signal given by O. Specifically,

Pr(\omega=\overline{\omega} | s = h, \rho=c) = q_{h}^{c}
Pr(\omega = \overline{\omega} | s = h, \rho = e) = q_{h}^{e}
Pr(\omega = \overline{\omega} | s = l, \rho = c) = q_{l}^{c}
Pr(\omega = \overline{\omega} | s = l, \rho = e) = q_{l}^{e}

There several general equilibria in the game; however I am only focusing on one (which I think is the most interesting) semi-separating equilibrium, which I call Opposing Victory.

Opposing Victory

Proposition

There exists a semi-separating equilibrium where the Opposition party signals they want to end the war (\rho = e) with probability \frac{\beta}{\overline{\iota} \ \overline{\omega}} if \omega = \overline{\omega} and signals that they want to end the war (\rho = e) if the actual state of the war is not worth fighting (\omega = \underline{\omega}). Thus, the Opposition is semi-separating when the true state of the war is worth fighting. Moreover, the Opposition Party’s behavior is irrespective of the signal the Media sends (i.e. s = h or s = l). The Electorate retains the Incumbent iff their belief that the true state of the war is going well (q_{h}^{e} \geq \hat{q^{e}}) when the Media says it is going well (s = h) and the Opposition Party wants to end the war (\rho = e); iff the their belief that the true state of the war is going well (q_{l}^{e} \geq \hat{q^{e}}) when the Media says it is going poorly (s = l) and the Opposition Party wants to end the war (\rho = e); and with certainty if the Opposition Party wants the was to continue (\rho = c), irrespective of the Media’s signal (s).

Because O values the national interest at a rate of \iota \sim U[0, \overline{\iota}], we can find the probability that \iota is above or below \frac{\frac{\beta}{\omega}}{\overline{\iota}}, which is the same as \frac{\beta}{\overline{\iota} \ \overline{\omega}}.

Assuming that O and E abide by the strategies outlined above, then Bayes’ rule tells us that,

Pr(\omega = \overline{\omega} | s = h,\ \rho = c) =

q_{h}^{c} = \frac{q (1 - \frac{\beta}{\overline{\iota} \ \overline{\omega}})(\alpha)}{q (1 - \frac{\beta}{\overline{\iota} \ \overline{\omega}})(\alpha) + 0 (1-q)(1 - \alpha)} \Rightarrow q_{h}^{c} = 1

and

Pr(\omega = \overline{\omega} | s = l,\ \rho = c) =

q_{l}^{c} = \frac{q (1 - \frac{\beta}{\overline{\iota} \ \overline{\omega}})(1 - \alpha)}{q (1 - \frac{\beta}{\overline{\iota} \ \overline{\omega}})(1 - \alpha) + 0 (1-q)(\alpha)} \Rightarrow q_{l}^{c} = 1

and

Pr(\omega = \overline{\omega} | s = h,\ \rho = e) =