Brock, W.A.

Zeeuw, A.J. de

English

Research Papers in Economics

 
 
The Repeated Lake Game
1
 
 
 
William A. Brock 
Social Systems Research Institute 
University of Wisconsin-Madison 
 
Aart de Zeeuw 
Department of Economics and CentER 
Tilburg University
2 
 
 
Abstract 
In common property ecological systems such as shallow lakes, an increasing number of users 
leads  to  additional  Nash  equilibria  with  considerably  lower  welfare.  When  the  game  is 
repeated, these bad outcomes can be turned into an advantage. 
 
 
JEL classification codes: Q20, C70 
Key words: ecological systems, shallow lakes, repeated games, trigger strategies  
 
                                                           
1This paper was initiated at a meeting of the Resilience Network that had financial support from the MacArthur 
Foundation.W.A. Brock thanks the Vilas Trust and the NSF under Grant Number SES 9911251 for essential 
financial support. None of the above are responsible for any errors or shortcomings in this paper. 
2P.O. Box 90153, 5000 LE Tilburg, the Netherlands, phone: +31-13-4662065, fax: +31-13-4663042, e-mail: 
A.J.deZeeuw@kub.nl 1. Introduction 
 
Lakes provide a variety of services such as water for irrigation, drinking and industrial uses, 
fishery, recreation and aesthetic enjoyment. But lakes are also used for dumping pollutants 
such as phosphorus from agricultural activities. These services are conflicting in the sense 
that the release of too many pollutants makes the lake useless in the other services. Especially 
shallow lakes have been studied intensively and it has been shown that due to internal loading 
processes, a sudden loss of ecological services may occur which is very difficult to restore 
(Carpenter  and  Cottingham,  1997;  Scheffer,  1997).  Lakes  are  usually  also  open-access 
resources. If a lake is used by a number of communities, open access implies the lake is not 
managed efficiently. In a Nash equilibrium the communities will release more phosphorus 
than they would in case of joint management of the lake, with a welfare loss as the result of 
that. 
It has been shown that the internal loading processes in the lake also affect the number of 
Nash equilibria of this game (Mäler, Xepapadeas and de Zeeuw, 2000). More specifically, for 
a low number of communities the Nash equilibrium is unique in their model, with a modest 
welfare loss, but when the number of communities is increased, a second Nash equilibrium 
occurs with a high loss in welfare. This result is worrisome at first sight but we like to argue 
in this paper that it may also be turned into an advantage. 
Remember  that  the  folk  theorem  in  repeated  games  teaches  us  that  cooperation  can  be 
sustained by an equilibrium in trigger strategies if the discount factor is high enough. The 
basic idea is that the communities stick to efficient loadings of phosphorus under the threat 
that deviation triggers the repeated static Nash equilibrium with lower welfare. The higher 
that loss of welfare, the lower is the minimal discount factor for which cooperation can be 
sustained by trigger strategies. An increase in the number of communities that use the lake 
has many effects. Welfare per community under optimal management changes as well as 
welfare in the static Nash equilibrium and welfare in case of cheating but as long as the static 
Nash equilibrium is unique, the minimal discount factor will not change much. However, 
when the second static Nash equilibrium occurs, this one will be used as trigger threat point 
and a non-monotonic jump in the minimal discount factor will occur. A similar result was 
found in the effect of changing numbers of firms on cartel formation in repeated Bertrand 
oligopolies with capacity constraints (Brock and Scheinkman, 1985). 
 
2. The lake model  
The  most  important  indicator  for  the  pollution  of  lakes  is  the  amount  of  phosphorus 
sequestered  in  algae.  The  essential  dynamics  of  the  pollution  process  (called  lake 
eutrophication) is given by (see Carpenter, Ludwig and Brock, 1999) 
where P is the amount of phosphorus in 
algae,  L  (for  loading)  is  the  release  of 
phosphorus  on  the  lake,  s  is  the  rate  of 
loss  consisting  of  sedimentation,  outflow  and  sequestration  in  other  biomass,  r  is  the 
maximum rate of internal loading and m is the anoxic level (for an extensive treatment of the 
lake model see Carpenter and Cottingham, 1997, or Scheffer, 1997). It is assumed that the 
lake dynamics is much faster than the changes in the agriculture activities that are responsible 
for the release of phosphorus. It follows that the loading L can be treated as a parameter: each 
level L leads to a steady-state level P for the differential equation (1). 
By substituting x = P/m, a = L/r, b = sm/r and by changing the time scale to rt/m, equation (1) 
can be rewritten as 
For most lakes it is found that ½ < b < 3 
3/8.  Figure  1  depicts  for  b  =  0.6  the 
steady-states x of differential equation (2) as a function of the loading parameter a. If the 
loadings a are increased from 0, at a certain point the steady-state phosphorus level x jumps 
to a high level which explains the sudden loss of ecological services. It is said that the lake 
flips from an oligotrophic state to a eutrophic state. If the loadings a are gradually decreased 
again, the lake remains eutrophic for a while before it flips back to an oligotrophic state 
(hysteresis effect). 
Mäler, Xepapadeas and de Zeeuw (2000) consider the situation that n communities have open 
access  to  the  lake  and  introduce  welfare  indicators  ln  ai  -  cx
2,  i  =  1,...,n,  c  >  0.  These 
indicators reflect the benefits of loading and the damage of accumulated pollution. Optimal 
management of the lake requires us to solve 
It is easy to show that the stationary points 
are given by 
(1)P dot (t) ~=~ L `-` sP(t) `+` r{P sup 2 (t)} 
over {P sup 2 (t) `+` m sup 2}, ~P(0) `=` P sub 0 
, 
(2)x dot (t) ~=~ a `-` bx(t) `+` {x sup 2 (t)} over 
{x sup 2 (t) `+` 1}, ~x(0) `=` x sub 0 . 
(3)maximize ~sum from {i=1} to n `ln `a sub i 
`-` ncx sup 2, ~s.t. ~ a `-` bx `+` {x sup 2 } over 
{x sup 2 `+` 1} ~=~ 0, ~a ~=~ sum from {i=1} 
to n `ln `a sub i . which is independent of n. For b = 0.6 and 
c = 2, figure 1 depicts the result: optimal 
management  leads  to  one  oligotrophic 
steady-state level of phosphorus. 
Mäler, Xepapadeas and de Zeeuw (2000) also show that the Nash equilibria, resulting from 
are given by 
More  precisely,  if  equation  (6)  has  one 
solution,  one  oligotrophic  Nash 
equilibrium  results,  and  if  it  has  three 
solutions, two Nash equilibria result, one 
oligotrophic and one eutrophic. It is easy 
to see graphically, in figure 1, what happens. Compare the right-hand sides of equations (4) 
and (6): increasing n means that the dashed curve is stretched out while the intersection 
points  with  the  x-axis  remain  in  place.  For  n  large  enough,  the  curves  have  two  more 
intersection points, of which the right one is a Nash equilibrium in the eutrophic area. Note 
that for n    , the series of oligotrophic Nash equilibria approaches the flip point. 
 
[Insert Figure 1 here] 
 
3. Trigger strategy equilibria 
 
Suppose now that the game in section 2 is infinitely repeated. In each period the communities 
choose phosphorus loadings, the lake adjusts and welfare levels result. Future welfare levels 
are  discounted  with  discount  factor      per  period.  Trigger  strategies  prescribe  that  the 
communities cooperate but switch to non-cooperative behavior when one of the communities 
deviates. The switch to the repeated static Nash equilibrium can be seen as a punishment that 
is credible since it is an equilibrium of the repeated game: trigger strategies are subgame 
perfect. If it does not pay to deviate unilaterally, trigger strategies are in equilibrium and 
support optimal management of the lake. This holds if (e.g. Friedman, 1989) 
where Wd denotes the highest welfare for 
a community if it deviates and the other 
communities  stick  to  cooperative 
behavior, Wc denotes the welfare per community in the Nash equilibrium for the stage game 
and Wo denotes the welfare per community per period under optimal management. It is, of 
(4)a ~=~ bx `-` {x sup 2 } over {x sup 2 `+` 1} 
~=~ 1 over c `(b over {2x} `-` 1 over {(x sup 2 
`+` 1) sup 2 }), 
(5)maximize  ~ln  `a  sub  i  `-`  cx  sup  2,  ~i `=` 
1,...,n, ~s.t. ~ a `-` bx `+` {x sup 2 } over {x sup 
2 `+` 1} ~=~ 0, 
(6)a ~=~ bx `-` {x sup 2 } over {x sup 2 `+` 1} 
~=~ n over c `(b over {2x} `-` 1 over {(x sup 2 
`+` 1) sup 2 }). 
(7)W sub d `+`   over {1 `-`  } `W sub c ~<~ 1 
over {1 `-`  } `W sub o ~ ~   ~>~ {W sub d `-` 
W sub o } over {W sub d `-` W sub c }, course,  possible  to  use  equilibria  in  more  sophisticated  strategies  and  achieve  stronger 
properties like renegotiation proofness, but we prefer to use the simplest framework to make 
our point. Welfare levels Wo and Wc are given by ln a0 - ln n - 2x0
2 and ln ac - ln n - 2xc
2, 
where (ao, xo) is the solution of equation (4) (for b = 0.6 and c = 2) and (ac, xc) is a solution of 
equation  (6).  If  the  number  of  communities  n  increases,  both  welfare  levels  Wo  and  Wc 
decrease by ln n but Wc decreases more because (ac, xc) moves away from the optimal point 
(ao, xo). More importantly, however, for n big enough (n > 3) two Nash equilibria occur, one 
oligotrophic  and  one  eutrophic.  Because  the  Nash  equilibrium  is  used  as  a  threat,  the 
eutrophic one is chosen, so that moving from n = 3 to n = 4 shows a significant drop in Wc. 
The numbers are given in table 1. It remains to derive Wd. Suppose that community i decides 
to deviate, the best it can do is described by 
which leads to 
It follows that Wd is simply given by ln aid 
-  2xd
2,  where  (aid,  xd)  is  the  solution  of 
equation (9) (for b = 0.6 and c = 2). The 
numbers  are  presented  in  table  1.  The 
welfare  level  Wd  decreases  but  not  as 
much  as  the  welfare  level  Wo  under 
optimal  management.  The  reason  is  that  a  large  number  of  cooperating  communities  is 
relatively beneficial for the one that deviates. 
 
[Insert Table 1 here] 
 
We are now ready to make our point. As can be seen from table 1, when n is increasing from 
2 to 3 and from 4 onwards, the minimal discount factor for which optimal management of the 
lake can be sustained by trigger strategies, increases gradually. However, when n increases 
from 3 to 4, this value drops dramatically. The occurrence of a second Nash equilibrium of 
the static game, with much lower welfare, threatens a much stronger punishment in case of 
deviation so that also for low discount factors, deviation is deterred. 
In order to keep our result transparent, we have chosen to present it in a simple framework 
with fixed parameter values. It is not so easy to derive the result analytically, because the 
dependence of the minimal discount factor  m, given by the right-hand side of inequality (7), 
on  the  number  of  communities  n  is  rather  complicated.  It  is  shown in figure 2 how the 
minimal discount factor  m depends on n if the parameter b is disturbed to b = 0.58 and b = 
(8)maximize ~ln `a sub i `-` cx sup 2 ~s.t. ~ a 
sub i `+` {n `-` 1} over n `a sub o `-` bx `+` {x 
sup 2 } over {x sup 2 `+` 1} ~=~ 0, 
(9)a sub i ~=~ bx `-` {x sup 2 } over {x sup 2 
`+` 1} `-` {n `-` 1} over n `a sub o ~=~ 1 over c 
`(b over {2x} `-` 1 over {(x sup 2 `+` 1) sup 2 
}). 0.62. The qualitative pattern remains the same. Numerical calculations for other parameter 
values show the same picture. 
 
[Insert Figure 2 here] 
 
4. Conclusion 
 
The folk theorem in repeated games teaches us that optimal management of a shallow lake, 
seen  as  repeated  game,  can  be  sustained  by  trigger  strategy  equilibria  for  high  enough 
discount factors. How high the discount factor should be depends on several factors such as 
how serious the threat is in case of deviation. Previous papers have shown that when the 
number of communities that use the lake is increased, additional Nash equilibria occur with 
low welfare. This paper employs that result to show that at some point a further increase in 
the number of communities leads to a large decrease in the minimal discount rate for which 
optimal management can be sustained. What may be seen as a negative result in the static 
context, can be turned into a positive one when the game is repeated.  
References 
 
Brock, W.A., Scheinkman, J.A. 1985. Price setting supergames with capacity constraints. 
Review of Economic Studies LII (3), 170, 371-382. 
 
Carpenter, S.R., Cottingham, K.L. 1997. Resilience and restoration of lakes. Conservation 
Ecology 1:2. 
 
Carpenter, S.R., Ludwig, D., Brock, W.A. 1999. Management of eutrophication for lakes 
subject to potentially irreversible change. Ecological Applications 9 (3), 751-771. 
 
Friedman, J.W. 1989. Game theory with applications to economics. Oxford University Press, 
New York. 
 
Mäler, K.-G., Xepapadeas, A., de Zeeuw, A. 2000. The economics of shallow lakes. CentER 
Discussion Paper 2000-69, Tilburg University. 
 
Scheffer, M. 1997. Ecology of shallow lakes. Chapman and Hall, New York. 
 
 
Table 1 
n  Wo  Wc  Wd    > 
1  -2.507       
2  -3.200  -3.214  -3.179  0.606 
3  -3.606  -3.635  -3.546  0.669 
4  -3.893  -7.514  -3.792  0.027 
5  -4.117  -8.656  -3.972  0.031 
  
Figure 1 The lake model under optimal management 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 2 The minimal discount factor as a function of the number of communities 
 
 
-0.05
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x
a
lake
0
0.2
0.4
0.6
0.8
1
2 3 4 5
b=0.58
b=0.60
b=0.62

Ecology of shallow lakes.

Game theory with applications to economics.

Management of eutrophication for lakes subject to potentially irreversible change.

Price setting supergames with capacity constraints.

Resilience and restoration of lakes.

The economics of shallow lakes.

The repeated lake game.

The repeated lake game.

Abstract

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