In this paper I illustrate by an example that strictly dominated strategies may affect the process of the equilibrium selection in coordination games. The strategy profile that gets selected may be both Pareto and risk dominated. This distinguishes it from the examples provided in Ellison (2000) and Maruta (1997).

Suren Basov

English

Research Papers in Economics

Equilibrium selection in coordination games: Why do
dominated strategies matter? 
Suren Basov
Melbourne University
Abstract
In this paper I illustrate by an example that strictly dominated strategies may affect the
process of the equilibrium selection in coordination games. The strategy profile that gets
selected may be both Pareto and risk dominated. This distinguishes it from the examples
provided in Ellison (2000) and Maruta (1997).
I am grateful to Murali Agastya, Peter Bardsley, and Andrew McLean for useful comments.
Citation: Basov, Suren, (2004) "Equilibrium selection in coordination games: Why do dominated strategies matter?."
Economics Bulletin, Vol. 3, No. 41 pp. 1−3
Submitted: October 28, 2004.  Accepted: October 28, 2004.
URL: http://www.economicsbulletin.com/2004/volume3/EB−04C70021A.pdf1 Introduction 
Coordination games characterize economic interactions in a large number of 
settings. The defining feature of such games is the existence of the multiple strict 
Pareto ranked Nash equilibria. The tools for equilibrium selection in such games are 
provided by the evolutionary game theory. The pioneering papers in this area are 
Foster and Young (1990), Kandori, Mailath, and Rob (1993) (henceforth, KMR), and 
Young (1993), which applied such models to 2×2 coordination games and showed 
that in the medium run players can coordinate on any strict Nash equilibrium, while in 
the long-run the risk-dominant outcome is selected as the unique stochastically stable 
solution. 
In this paper I investigate the sensitivity of the equilibrium selection results to 
the inclusion of strictly dominated strategies. Such an investigation is important, 
because the games we study in economics are usually stylised descriptions of real life 
strategic interaction, which leave out a lot of details. Therefore, the strategy set 
available to the players is sensitive to the modelling decisions.  
2 An Example 
In this Section I consider a simple coordination game and show that the 
presence of a strictly dominated strategy can affect the long-run outcome. Consider 
coordination game represented on Figure 1: 
 A  B  C 
A  2,2 0,0 c,0 
B  0,0 3,3 0,0 
C  0,c 0,0 0,4 
                                        Figure 1. 
Here c>4. This game has two pure strategy Nash equilibria (A,A) and (B,B), and a 
mixed strategy equilibrium (0.6A+0.4B, 0.6A+0.4B). Strategy C, on the other hand, is 
strictly dominated (for example, by strategy 0.9A+0.1B) and is weakly dominated by 
A. 
If one believes that strictly dominated strategies should not affect the outcome 
of the equilibrium selection process, the long-run prediction for this game should be 
the same as for the following 2X2 coordination game: 
 A  B 
A  2,2 0,0 B  0,0 3,3 
                                        Figure 2. 
However, as I will show shortly, the standard KMR dynamics selects different 
outcomes for these games, provided c>4.5. 
To describe the standard KMR dynamics assume that the population consists 
on N players and at period t=0 each player is characterized by a particular strategy she 
plays. The strategy choice of a player in the next period is the best response to the 
current population strategy profile with probability 1-ε , but with probability ε  the 
player suffers from noise, in which case the strategy is selected at random and all 
strategies are selected with positive probabilities. Noise occurs independently across 
both players and periods. If ε >0 the model described about possesses a unique steady 
state distribution. The limit of this distribution as ε   goes to zero is known as 
stochastically stable equilibrium.  
It is a well-known result that for the game depicted at Figure 2 the 
stochastically stable equilibrium is (B, B). The stochastically stable outcome of the 
game depicted at Figure 1, however, depend on the value of c. To see this let us 
denote by D(A) (D(B)) the basin of attraction of pure strategy A (B), i. e. the set of 
all mixed strategies to which A (B) is a best reply. These sets are illustrated on 
Figure 3.  The vertices represent monomorphic populations playing particular 
strategies. It is easy to check that strategies A and B earn the same payoffs against 
strategy profiles (3/5, 2/5, 0) and (0, c/(3+c), 3/(3+c)), where numbers in brackets 
represent the fractions of A, B, and C-strategists respectively.   
                                                       A 
                                   R(A) 
                            (3/5, 2/5, 0) 
                                                 D(A) 
                          
                                       D(B) 
                      
                 B                                                                  C 
                                  CR(A)       (0, c/(3+c), 3/(3+c)) 
Following Ellison (2000), define R(A), the radius of D(A), as the minimal 
distance from A to D(B) and CR(A), the coradius of D(A), as the minimal distance 
from B to D(A). Then from Figure 3 
  2R(A)=2/5, CR(A)=3/(3+c). 
As the radius exceeds the coradius A is the unique stochastically stable equilibrium. 
This happens for c>9/2.  Therefore, if an A-strategist fairs against a C-strategist 
sufficiently better than a B-strategist,  (A, A) rather then (B, B) will become the 
stochastically stable equilibrium. Therefore, the long-run outcome of a strategic 
interaction can be affected by the presence of a strictly dominated strategy. 
3 Conclusions 
  In this note I demonstrated by an example that the long-run equilibrium of a 
game may be sensitive to the presence of strictly dominated strategies. Indeed, players 
may coordinate on a strategy that is both Pareto and risk dominated, provided it fairs 
well against a strictly dominated strategy. The result is rather disturbing because it can 
be interpreted as the sensitivity to the modelling assumptions. Indeed, assume that 
players are firms. Let us interpret different strategies as R&D programmes, and 
suppose there is complementarity between the programmes of different firms. Since 
the number of different R&D programmes can be numerous and had to model 
explicitly, an economist conducting a study of the industry might be willing to 
concentrate only on viable alternatives, leaving others out. One definition of a viable 
alternative is that it is not strictly dominated. As we have seen, however, leaving out 
alternatives that are judged non-viable can affect the long-run prediction for the 
behaviour of the industry. 
References 
[1] Ellison, G., 2000, Basins of attraction, long-run stochastic stability, and the speed 
of step-step evolution, Review of Economic Studies, 67, 17-45. 
[2] Foster, D., Young, P., 1990, Stochastic evolutionary game dynamics, Theoretical 
Population Biology, 38, 219-232. 
[3] Kandori, M., Mailath, G., Rob, R., 1993, Learning, mutation and long run 
equilibria in games, Econometrica, 61, 29-56. 
[4] Maruta, T., 1997, On the relationship between risk-dominance and stochastic 
stability, Games and Economic Behaviour,19, 221-34 
[5] Young, P., 1993, The evolution of conventions, Econometrica, 61, 57-84. 
  3

Basins of attraction, long-run stochastic stability, and the speed of step-step evolution,

Learning, mutation and long run equilibria in games,

On the relationship between risk-dominance and stochastic stability,

Stochastic evolutionary game dynamics,

The evolution of conventions,

Equilibrium selection in coordination games: Why do dominated strategies matter?

http://www.accessecon.com/pubs/EB/2004/Volume3/EB-04C70021A.pdf

Equilibrium selection in coordination games: Why do dominated strategies matter?

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