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    {\bf The Horatii and the Curiatii}\footnote{Eric Rasmusen:

Professor of Business Economics

and Public  Policy and Sanjay Subhedar Faculty Fellow,   Indiana

University,

Kelley School of Business, BU 456,   

  1309 E 10th Street,

  Bloomington, Indiana, 47405-1701.

  Office: (812) 855-9219.   Fax: 812-855-3354.   Erasmuse@indiana.edu.

Php.indiana.edu/$\sim$erasmuse.} \\

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 September 21, 2000 \\

                    

                       

          

 	



  

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  A Roman, Horatius, unwounded, is fighting the three Curiatius

brothers from Alba, each of whom is wounded.  If Horatius continues

fighting, he wins with probability 0.1, and the payoffs are (10,-10)

for (Horatius, Curiatii) if he wins, and (-10,10) if he loses. With

probability $\alpha = 0.5$, Horatius is panic-stricken and runs away.

If he runs and the Curiatii do not chase him, the payoffs are (-20,

10).  If he runs and the Curiatius brothers chase and kill him, the

payoffs are (-21, 20). If, however, he is not panic-stricken, but he

runs anyway and the Curiatii give chase,   he is able to kill the

fastest brother first and  then  dispose of the other two, for payoffs

of (10,-10). Horatius is, in fact, not panic-stricken.





 \hspace*{16pt} (a) With what probability $\theta$ would the

Curiatii give chase if Horatius were to run? \\

    \hspace*{72pt} \underline{  {\it Answer.}} In a mixed-strategy

equilibrium,

 \begin{equation} \label{e76} 

 \pi_h(run) = \pi_h (not\; run), 

\end{equation} 

 so  

 \begin{equation} \label{e77} 

 \theta (10) + (1-\theta) (-20) = 0.1(10) + 0.9 (-10) 

 \end{equation} 

 and  

$    {\theta^* = \frac{12}{30} =0.4}$.





  \hspace*{16pt} (b) With what probability $\gamma$ does Horatius

run?\\

     \hspace*{72pt} \underline{  {\it Answer.}} In a mixed-strategy

equilibrium,

 \begin{equation} \label{e78} 

 \pi_c(chase) = \pi_c (not\; chase), 

\end{equation} 

 so  

\begin{equation} \label{e79} 

\begin{array}{l} 

 \alpha(20) + (1-\alpha) \gamma (-10) + (1-\alpha)(1-\gamma)[ 0.1(-10) 

+ 0.9 ( 10)] =\\ 

 \alpha(10) + (1-\alpha) \gamma (10) + (1-\alpha)(1-\gamma)[ 0.1(-10) + 

0.9 ( 10)],  

\end{array} 

 \end{equation} 

 which reduces to 

 \begin{equation} \label{e80} 

20 \alpha -10 \gamma + 10\alpha \gamma= 10\alpha + 10\gamma -10\alpha 

\gamma, 

 \end{equation} 

 so $\gamma^* = \frac{\alpha}{2-2\alpha}$, which equals {  \rm 0.5} if

$\alpha=0.5$. 



  \hspace*{16pt} (c) How would $\theta$ and $\gamma$ be affected

if the Curiatii falsely believed that the probability of Horatius

being panic-stricken was 1?  What if they believed it was 0.9? \\

   \hspace*{72pt} \underline{  {\it Answer.}} If $\alpha=1$ (or if the

Curiatii think

$\alpha=1$),   there is no mixed-strategy equilibrium, because $

\pi_c(chase) > \pi_c (not\; chase)$ under all possible circumstances.

Thus, the equilibrium is $\gamma^*=1$, $\theta^*=1$: Horatius runs

and the Curiatii chase. If $\alpha =0.9$, it is still true that $

\pi_c(chase) > \pi_c (not\; chase)$ even if $\gamma=1$, so the

equilibrium is the same as if $\alpha =1$. 





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