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\noindent
April 21, 2011   Eric Rasmusen, Erasmuse@indiana.edu.
 Http://www.rasmusen.org.    

\begin{LARGE} 
\begin{center} 
{\bf  The Codfish    Game  }
 \end{center}
 \end{LARGE}

 

   Each of eight countries   in a fishery decides how many fish to catch each decade. Each country  $i$ picks  an integer  number $X_{it}$ as its fishing catch for  decade $t$. The country's profit for    decade  $t$ is 
\begin{equation}
20X_{it} - X_{it}^2.
\end{equation}
     Thus,   diminishing returns set in after  a certain point and the marginal cost is too high for further fishing to be profitable.

	 The fish population starts at 112 (14 per country)  and the game continues for 5 decades. Let $Q_1$ denote the fish population at the start of decade 1. In decade 2, the population is 
\begin{equation}
1.5 \cdot ( Q_1 - (X_{1t} + X_{2t} + X_{3t} +...)),\; rounded\; up, 
\end{equation} 
 up to a maximum of 400,  where $X_{it}$ is country $i$'s catch in decade $t$.


   If  $ X_{11} = 30$ and $X_{21} =X_{31}=...=X_{81}= 3$, then the first country's profit is $20*30-30^2 = 600-900 = -300$, and each other country earns $20*3-3^2 = 60-6 = 54$.   The second-decade fish population would be $Q_2= 1.5*(112-30 - 7[3])= 1.5(82-21) = 1.5(61) = 92$.

 \bigskip


(1) Britain is the only country. 

 

(3) Start over with 112 fish. The   eight countries choose independently. Each country writes down its catch  on a  piece  of paper, which it hands in to the instructor. The instructor opens them   as he receives them.  If the attempted catch exceeds the total fish population, those countries which handed in their catches first get priority, and a country's payoff is $(20Z_{it} - X_{it}^2)$, where $Z_t$ is its actual catch and $X_t$ is its attempted catch, what it wrote down.   Do this for 5 decades. 

 (2)  Start over with 112 fish. An International Codfish Commission    chooses the catch for all eight countries. Each country gets to propose a per-country catch quota  to the  ICC, so the total catch will be 8 times that quota.  The proposals will be discussed publicly,and the Commissioner will decide.     Once  the catch is finalized, the instructor calculates the next year's fish population, and  the quota process will be repeated. 

(4) Repeat  the entire process but this time allow any countries that so wish to form a binding treaty and submit their catches jointly, on one piece of paper.


 

\newpage

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\begin{small}

\begin{center}
{\bf  Scoresheet for ``Fisheries''}
\end{center} 

\noindent
Your Name: 

  Your Country's Name:\\
 
 \vspace{48pt}

 \hspace*{-48pt}
\begin{tabular} {| l ||  l | l | l  ||    l | l | l  ||   l | l | l  ||  }
\hline
   &  &    &    &  &    &   &  &    &     \\
Decade &  Fish      & Your     &  Your  &  Fish      & Your       &  Your    &  Fish      & Your      &  Your  \\
   & Population &   Catch  &  Payoff   & Population &   Catch  &  Payoff & Population &   Catch  &  Payoff\\
   &  &    &    &  &    &   &  &    &     \\
\hline
1   &  &    &   &  &    & &  &    &   \\
   &  &    &    &  &    &   &  &    &     \\
\hline
2   &  &    &    &   &  &    & &  &       \\
     &  &    &    &  &    &   &  &    &     \\
\hline
3   &  &    &   &   &  &    & &  &        \\
    &  &    &    &  &    &   &  &    &     \\
\hline
4   &  &    &    &   &  &    & &  &        \\
    &  &    &    &  &    &   &  &    &     \\
\hline
5   &  &    &   &   &  &    & &  &       \\
    &  &    &    &  &    &   &  &    &     \\
\hline
\hline
Total &     &    & &   &  &    & &  &     \\
  Payoff  & -----  &  -----    &    & -----  &   ----- &
& -----  & -----  &     \\
   &  &    &    &  &    &   &  &    &     \\
\hline
\end{tabular} 

 \end{small}
 
% \end{LARGE}
  
\newpage

 


\begin{tabular} {  |l|| l| l| l | l| l|  }
\hline
     &    1 \hspace*{12pt}  &     2 \hspace*{12pt} &   3\hspace*{12pt} & 4 \hspace*{12pt}& 5\hspace*{12pt}  \\
\hline
 \hline
INITIAL POPULATION &  &    &    &   &      \\
 \hline
\hline
USA   &  &    &    &     &      \\
\hline
 Canada&  &    &    &   &       \\
\hline
Korea&  &    &    &  &    \\
\hline
 Japan&  &    &    &    &        \\
\hline
Britain&  &    &    &     &      \\
\hline
Russia&  &    &    &    &        \\
\hline
 France&  &    &    &     &       \\
\hline
 China&  &    &    &     &    \\
\hline
\hline
TOTAL CATCH &  &    &    &   &       \\
\hline
\end{tabular}



\vspace*{24pt}

 



\begin{tabular} {  |l|| l| l| l | l| l|  }
\hline
     &    1 \hspace*{12pt}  &     2 \hspace*{12pt} &   3\hspace*{12pt} & 4 \hspace*{12pt}& 5\hspace*{12pt}  \\
\hline
 \hline
INITIAL POPULATION &  &    &    &   &      \\
 \hline
\hline
USA   &  &    &    &     &      \\
\hline
 Canada&  &    &    &   &       \\
\hline
Korea&  &    &    &  &    \\
\hline
 Japan&  &    &    &    &        \\
\hline
Britain&  &    &    &     &      \\
\hline
Russia&  &    &    &    &        \\
\hline
 France&  &    &    &     &       \\
\hline
 China&  &    &    &     &    \\
\hline
\hline
TOTAL CATCH &  &    &    &   &       \\
\hline
\end{tabular}

\vspace*{24pt}


 


\begin{tabular} {  |l|| l| l| l | l| l|  }
\hline
     &    1 \hspace*{12pt}  &     2 \hspace*{12pt} &   3\hspace*{12pt} & 4 \hspace*{12pt}& 5\hspace*{12pt}  \\
\hline
 \hline
INITIAL POPULATION &  &    &    &   &      \\
 \hline
\hline
USA   &  &    &    &     &      \\
\hline
 Canada&  &    &    &   &       \\
\hline
Korea&  &    &    &  &    \\
\hline
 Japan&  &    &    &    &        \\
\hline
Britain&  &    &    &     &      \\
\hline
Russia&  &    &    &    &        \\
\hline
 France&  &    &    &     &       \\
\hline
 China&  &    &    &     &    \\
\hline
\hline
TOTAL CATCH &  &    &    &   &       \\
\hline
\end{tabular}
\vspace*{24pt}
\begin{center}
 {\bf  TABLE 2:    Histories }
\end{center}



\newpage
 
 

 \bigskip 
\noindent
 {\bf Instructor's Notes   }

Equipment: Scoresheets\\

  

  This  game illustrates the common-pool resource problem. It is  a variant on the Prisoner's Dilemma.

 I have also tried starting the game with 230 fish, to show how the number would shrink, stopping it after a few rounds to note that eventually the number would drop to zero. 

 I usually haven't tried the last variant, the one where they can form binding agreements. 

 

  

 \noindent 
  (a) If the social planner wants the catch to be $Z$, it should spread that $Z$ evenly across the 8 countries so $X=Z/8$, since the cost of catching fish, $X^2$, is convex, so the marginal cost, $MC=2X$, is increasing.

 
\noindent
(b) If there were just one decade, the social planner would solve 
 \begin{equation}
\stackrel{Maximize} {X}  \;\; 8(20X - X^2), 
\end{equation}
which has the first-order condition
 $$
160-16X = 0, 
$$
so  $X^*=10$.  If we denote the total catch by $Z_t$ in decade $t$, that means if there is just one decade then $Z^*=80$.   Beyond $X=10$, the marginal cost exceeds the marginal benefit, so even if the fish population were higher than 80, it would be  inefficient to catch more than 10 fish per country. 

 
 \noindent
 (c)  
  Suppose there is no discounting and 5 periods. In this case, the aim is to choose a harvest path that maximizes the total catch over the 5 decades. This path balances two things: (i) waiting to catch fish later results in a larger population, and (ii) it is never efficient to catch more than 80 fish per decade. 

 The optimal policy will  include $Z_5=80$, catching all the fish in decade 5, since we don't care about any future periods. That is artificial, of course. 


\bigskip 
 \noindent
(d) Suppose there is no discounting and an infinite number of periods.  
 It turns out that the optimum is to reach a ``sustainable'' level of fishing. The aim will be to get the fish population up to a level where a harvest of 80 fish per decade can be sustained forever.  That requires a population of 240, so  that 240 = 1.5(240-80).   One policy is to catch no fish until that population is reached. Another  is to approach 160 gradually. 

 
Note that ``sustainable'' does not mean ``maximum sustainable''. If the fish population were to start above 240, it should immediately be fished down to stay at 240. 

    

 \newpage

These tables show some spreadsheet calculations (see \url{http://rasmusen.org/GI/games/01_fish.xls}).

 \noindent
{\bf The Optimum over Just Five Decades} (approximated, because of rounding)


\begin{tabular}{	llllllll}	
Year	&	Fish	&	Catch	&	End Fish	&	cost of catch	&	Benefit of catch	&	Net benefit	&	Total benefit	\\
1	&	112	&	26	&	129	&	84.5	&	520	&	435.5	&	3341.25	\\
2	&	129	&	44	&	127.5	&	242	&	880	&	638	&		\\
3	&	127.5	&	55	&	108.75	&	378.125	&	1100	&	721.875	&		\\
4	&	108.75	&	63	&	68.625	&	496.125	&	1260	&	763.875	&		\\
5	&	68.625	&	68	&	0.9375	&	578	&	1360	&	782	&		\\
\end{tabular}

\vspace*{1in}
 \noindent
{\bf
Getting to a Sustainable Catch Quickly}

 \begin{tabular}{	llllllll}	
Year	&	Fish	&	Catch	&	End Fish	&	cost of catch	&	Benefit of catch	&	Net benefit	&	Total benefit	\\
1	&	112	&	0	&	168	&	0	&	0	&	0	&	2552	\\
2	&	168	&	8	&	240	&	8	&	160	&	152	&		\\
3	&	240	&	80	&	240	&	800	&	1600	&	800	&		\\
4	&	240	&	80	&	240	&	800	&	1600	&	800	&		\\
5	&	240	&	80	&	240	&	800	&	1600	&	800	&		\\
	\end{tabular}

\vspace*{1in}
 \noindent
{\bf
Getting to a Sustainable Catch Slowly }

\begin{tabular}{	llllllll}	
Year	&	Fish	&	Catch	&	End Fish	&	cost of catch	&	Benefit of catch	&	Net benefit	&	Total benefit	\\
1	&	112	&	0	&	168	&	0	&	0	&	0	&	2789.5	\\
2	&	168	&	40	&	192	&	200	&	800	&	600	&		\\
3	&	192	&	50	&	213	&	312.5	&	1000	&	687.5	&		\\
4	&	213	&	52	&	241.5	&	338	&	1040	&	702	&		\\
5	&	241.5	&	80	&	242.25	&	800	&	1600	&	800	&		\\
\end{tabular}

 
 

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