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\noindent
22 February 2006. 5 March 2007.   Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org.   Comments    welcomed.
 
  
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\begin{center}
 {\bf  The War of Attrition Game} 
 \end{center}
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\bigskip
\noindent
{\bf The Rules of the Game} 

   Each firm consists of 3 students. Each year a firm must decide
whether to stay in the industry or to exit. If it stays in, it incurs
a fixed cost of 300 and a marginal cost of 2,  and it chooses an
integer  price
at which to sell (0,1,2,3,4,5,6,7,8,9, 10).  The firms can lose unlimited amounts of money; they
are backed by large corporations who will keep supplying them with
capital indefinitely.


Demand is inelastic at 60 up to a threshold price of
\$10/unit,  above which the quantity demanded  falls to zero.

 Each firm writes down its price (or the word ``EXIT'')  on a piece of
paper and gives it to
the instructor. The instructor then writes the strategies of each firm
on the blackboard (EXIT or price),   and
the firms charging the lowest price split the 60 consumers evenly. 

The game then starts with a new year, but any firm that has exited is
out permanently and cannot re-enter. The game continues until only one
firm is active, in which case it is awarded a prize of \$2,000, the
capitalized value of being a monopolist.  This means the game can
continue forever, in theory. The instructor may wish to cut it off at
some point, however.

 The game can then be restarted and continued for as long as class
time permits.
	
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\noindent
 \hspace*{-72pt} Your Firm:   \hspace*{96pt}  Your Names:   
 
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\begin{tabular} {| l | l| l| l | l| l| l |l| }
\hline
   &  &    &   &  &   &   &  \\
Year  & No.&  Price  & Winning  &  Revenue  &     Costs    & Current &
Cumulative \\
  &  of   Firms & or EXIT   & Price  &  (0 or PQ)  &  (0,  300, or 300+2Q)    &  Profit &
Profit \\
 &  &    &   &  &   &   &  \\
\hline
1  &  &    &   &  &   &   &  \\
\hline
2  &  &    &   &  &   &   &  \\
\hline
 3 &  &    &   &  &   &   &  \\
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\end{tabular} 

 


\begin{center}
 {\bf  TABLE 1:  SCORESHEET }
\end{center}

No. of Firms is the number of firms that have not yet exited. Winning Price is the lowest price charged that year.    If your firm
exits in period $n$, it does not pay the fixed cost that period.

 

Cumulative Profit is the sum of the firm's  profits in that industry
up to and including that period and the prize of 2,000 if your firm is the survivor. It starts at zero again when only one
firm is left in the industry and the game starts in a new industry.
 

% \end{LARGE}



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 \bigskip
\noindent
{\bf  Notes on the War of Attrition}

 There are several ways to vary the game. I suggest starting with a policy that firms cannot discuss strategies with each other. 


 The instructor may wish to halt the game temporarily after a few
rounds and have public discussion.

 The main point of the first play of the game will be that even the surviving firm  might end up with a negative payoff, because the game may continue past 7 rounds. 



Students at first may think this is the same as ordinary repeated
oligopoly, and that their   goal  is to achieve
collusion, which they know is difficult. If you next change the rules to allow discussion between firms,  they will realize that collusion doesn't solve their problem.  

  This game will  probably take 40 to 75 minutes.  It is flexible because you can choose how many times to repeat it. 



 

\noindent
 LESSONS: 

  1. In equilibrium, exiting or staying in   have equal expected
payoffs, if
everyone is risk neutral. Staying in  is
riskier.

2. If you can  convince  your rivals  that you are not going to exit,
your payoff will be higher.

3. Profits are negative and price is near  marginal cost   until only
one firm
survives.

4. You might like to charge less than 2, just to inflict losses on
your rivals. If you charge 2, you will earn zero profits (ignoring
fixed costs). You might like to charge 1 or 0, though. The purpose
would be to persuade your rivals that you will stay in forever,
despite losses, because you are malicious,  stupid, or insane (Nixon's
``Madman Theory'').

5. The game can go on more than the  7 periods which create a loss of
2,100, more than the value of the prize. Sunk  costs are sunk.

6. This is like  Shubik's ``dollar auction'', except you can get out
for
free at
any time.

7.  The only  symmetric Nash Equilibria are in mixed strategies.
There are lots of asymmetric Nash equilibria, however-- for example,
that Firm 1 stays in no matter what, but other firms exit immediately.

 8. It does no  good to collude. The colluding firms would just keep
losing money.  If two firms collude, for example, they would choose a
price of 10, for revenues of (10-2) (6) = 480 excluding fixed costs.
But their fixed costs would be 2*300=600.


9.  It is worth discussing what happens if some firms have limited
budgets-- that is, if they must exit after losing, say, 1,000. Those
firms ought to exit immediately-- if other firms are rational, the
other firms will just wait them out.
  
 
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