Coordination and culture

Abstract

Culture constrains individual choice, rendering certain actions impermissible or taboo. While cultural constraints may regulate behavior within a group, they can have a pernicious effect in multicultural societies, inhibiting the emergence of unified social conventions. We analyze interactions between members of two cultural groups who are matched to play a coordination game with an arbitrary number of actions. Due to cultural constraints, miscoordination prevails despite strong incentives to coordinate behavior. In an application to identity-based conflict, exclusive ethnic and religious identities persist in poorer and more unequal societies. Occasional violation of cultural constraints can make miscoordination even more stable.

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Notes

  1. 1.

    For example, Tetlock et al. (2000) find that experimental subjects express moral outrage at even contemplating certain taboo transactions, including buying and selling of human body parts for medical transplant operations, surrogate motherhood contracts, adoption rights for orphans, votes in elections for political office, the right to become a US citizen and sexual favors. See Roth (2007) for an account of how repugnance constrains markets.

  2. 2.

    See also Henrich et al. (2001) who suggest that culture “limits choice sets” [p. 357]. Conventional definitions in economics equate culture with shared preferences (e.g., Bisin and Verdier 2000) and strategic beliefs (e.g., Greif 1993).

  3. 3.

    This phenomenon is qualitatively different to the selection of a Pareto-inefficient coordination equilibrium when agents use the same action set (Kandori et al. 1993; Young 1993a). Here, play might not settle into any coordination equilibrium, even though every coordination equilibrium Pareto dominates miscoordination.

  4. 4.

    Important contributions to understanding identity-based conflict and cooperation are also made by Fearon and Laitin (1996), Laitin (2007), Esteban and Ray (2008), McBride et al. (2011), Jha (2013) and Sambanis and Shayo (2013). Sambanis and Shayo present a formal theory in which individuals either identify with their ethnic group or the nation. Under certain conditions, multiple equilibria exist and cascades of ethnic identification can occur (see also Kuran 1998). Such miscoordination between cultural groups, as we show, is a more general phenomenon, not limited to ethnic conflict and independent of specific feedback mechanisms. By analyzing an explicit out-of-equilibrium process, we are also able to select among different equilibria and show that miscoordination is a surprisingly stable outcome.

  5. 5.

    A notable exception is Young (1993b) who examines bargaining between members of heterogeneous groups.

  6. 6.

    On robustness of these results to different specifications of revision opportunities and tie-breaking rules, see Alós-Ferrer and Netzer (2014).

  7. 7.

    In addition, when agents are differentially located in space, different groups can settle on different conventions (e.g., Young and Burke 2001; Blume 2003; Ioannides 2006).

  8. 8.

    Members of ethnic or religious minorities often adopt different modes of behavior when interacting outside of their communities (see Akerlof and Kranton 2000, pp. 738–739).

  9. 9.

    Over time, we can think of players “dying” and their roles being filled by incoming players who inherit their predecessor’s culture. One such example is the vertical transmission of culture from parent to child.

  10. 10.

    The results in this section hold if we assume that in every period each agent is matched with every agent from the other group, with a set \(R_t\) of individuals selected to revise their strategies as above. The application analyzed in Sect. 3.1, however, is developed for pairwise interactions.

  11. 11.

    For theories of veiling see Patel (2012) and Carvalho (2013).

  12. 12.

    Recall that state \(z'\) is accessible from z if there exists a positive probability path from z to \(z'\). The states communicate if they are each accessible from the other.

  13. 13.

    A set of states E is closed if for all \(x\in E\) and \(y\notin E, P_{xy} =0\). A recurrent class is a closed communication class.

  14. 14.

    This does not mean that the state space can be partitioned into a basin of attraction for each state of coordination and a basin of attraction for \(\mathcal {M}\). There are some states from which both a state of coordination and miscoordination can be reached with positive probability, depending on which players are chosen to revise their strategy.

  15. 15.

    This accords with empirical evidence. For example, interracial, interethnic and interreligious marriages are more likely to end in divorce than homogamous marriages (see Xuanning 2006 and references therein) and cultural differences reduce the volume of and returns from cross-border mergers (Ahern et al., forthcoming). The subdiscipline of cross-cultural management is devoted to such issues. See Akerlof and Kranton (2010) for numerous examples of organizational conflict along gender, class and racial lines. On the other hand, Page (2007) shows how organizations can benefit from diversity.

  16. 16.

    The population size n needs to be sufficiently large only to break ties in resistances between recurrent classes induced by rounding up to the nearest integer. When n is small, multiple classes can be selected. But as long as \(\mathcal {M}\) is stochastically stable for some population size n it remains stochastically stable for all smaller n.

  17. 17.

    Naturally, the results coincide when \(|X_A \cap X_B|=1\), but our analysis is more general.

  18. 18.

    In Kajii and Morris’ terminology, what we define is a \((\tfrac{1}{2},\tfrac{1}{2})\)-dominant equilibrium.

  19. 19.

    Note that individuals are born into either group A or B. This is not a choice. However, they can choose whether to emphasize their exclusive group identity or adopt an inclusive identity.

  20. 20.

    This contest function was first applied to conflict by Skaperdas (1992) and intergroup conflict by McBride et al. (2011) and Sambanis and Shayo (2013). Sambanis and Shayo suggest that lower levels of ethnic conflict are observed when a common national identity is adopted, because individuals who adopt an ethnic identity aim to maximize the share of national resources captured by their ethnic group.

  21. 21.

    Of course, the value of access to group resources \(\beta \) may depend on w. But we have reason to think that this effect is weak. Production of club goods typically involves labor-intensive modes of production (Iannaccone 1992; Berman 2000), especially in poorer societies (see Wickham 2002).

  22. 22.

    Fearon and Laitin (2003) report a large and significant negative relationship between income levels and the incidence of civil war. Their explanation centers on the link between higher income levels and the opportunity cost of insurgency. More closely related to our explanation is work by McBride et al. (2011). They propose that conflict destroys resources. Hence in a repeated games setting, higher income levels provide greater incentive for groups to strike a peaceful agreement.

  23. 23.

    A nonvanishing likelihood of violating cultural constraints is analytically equivalent to allowing individuals to revise their cultural constraints.

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Acknowledgments

This paper has benefitted greatly from advice by Peyton Young, as well as comments by the Associate Editor, two anonymous referees, Blake Allison, Ken Binmore, Rob Boyd, Michael Caldara, David Myatt, Tom Norman, Michael Sacks, Stergios Skaperdas, Christopher Wallace and seminar participants at UCLA Anderson, Claremont Graduate University, the Institute for Mathematical Behavioral Sciences, UC Irvine and the University of Western Australia. All errors are mine. Financial support from the Commonwealth Bank Foundation in the form of a John Monash scholarship is gratefully acknowledged.

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Correspondence to Jean-Paul Carvalho.

Appendix

Appendix

Proof of Proposition 1.

Proof

We shall characterize the set of recurrent classes of the best response dynamic. As any finite Markov chain converges almost surely to one of its recurrent classes, this establishes the proposition.

Firstly, we claim that there are no recurrent classes of the best response dynamic other than the states of coordination and \(\mathcal {M}\). To establish the claim it suffices to show that there is a positive probability of transiting from any state to a state of coordination or a state in \(\mathcal {M}\) in a finite number of periods.

Consider an arbitrary time t and state \(z^t\). Let x be a best response for group A members. Suppose that in period t, the set of revising players is \(R_t=N_A\) and that each of them chooses x. In addition, suppose that in period \(t+1, R_{t+1}= N_{B}\). All of this occurs with positive probability.

Case 1 \(x\in X_{A}\cap X_{B}\). (By definition, this is impossible if the cultures are incompatible.) Then the expected payoff to each revising group B member from playing x is \(\theta _{Bx} + \delta _{Bx}\) for all samples. The payoff from choosing \(x'\ne x\) is at most \(d_{B}\), which is less than the payoff from x by \(\textit{ND}\). Hence each revising group B member chooses x and the dynamic transits with probability to a state of coordination in two periods.

Case 2 \(x\notin X_{A}\cap X_{B}\). Then the payoff to each revising group B member from playing any \(x'\in X_{B}\) is \(\delta _{Bx'}\). Hence each revising group B member chooses an action in \(\widetilde{X}_{B} \equiv argmax_{x\in X_{B}}\delta _{Bx}\). When the cultures are misaligned, every such action lies outside \(X_{A}\cap X_{B}\), as does x. Hence the dynamic transits with positive probability to a state in \(\mathcal {M}\) in two periods. When the cultures are aligned, we have either \(\tilde{X}_A\cap X_B\) is nonempty or \(\tilde{X}_B\cap X_A\) is nonempty. Thus the dynamic transits with positive probability in at most two periods to a state in which each group B member plays an action \(x'\in X_{A}\cap X_{B}\). From there, the argument used in case 1 establishes that the dynamic transits to a state of coordination in finite time.

This establishes the claim.

The fact that the states of coordination are recurrent classes of the best response dynamic if and only if the cultures are compatible follows immediately from the argument in case 1. That \(\mathcal {M}\) is a recurrent class if and only if the cultures are misaligned follows immediately from the argument in case 2. This establishes the proposition.

\(\square \)

Proposition 2. The proof of Proposition 2 employs two tree-surgery arguments. These arguments make use of the following three lemmas and corollary, which compare the resistance of transitions between recurrent classes, defined as follows. The cost of a path \((z^1,z^2,\ldots z^H)\) is the minimum number of errors required for the perturbed dynamic (\(\varepsilon > 0\)) to transit from state \(z^1\) to \(z^H\) along this path. The resistance \(r(z^{1},z^H)\) is the minimum cost of a path starting at \(z^1\) and ending in \(z^H\). The definition can be extended so that for sets E and \(E'\), the resistance \(r(E,E')\) is the minimum cost of a path starting in some state in E and ending in some state in \(E'\).

In addition, let \(x_i(z)\) be the action played by i in state z. A direct path \((z^1,z^2,\ldots z^H)\) from state \(z^1\) to \(z^H\) is one in which \(x_i(z^h)\in \{x_i(z^1), x_i(z^H)\}\) for all \(h=1,2, \ldots H\) and \(i\in N\).

Define \(X_i(E) \equiv \{x_i(z): z\in E\}\). A direct path \((z^1,z^2,\ldots z^H)\) from set E to \(E'\) is one in which \(z^1\in E, z^H\in E'\) and \(x_i(z^h)\in X_i(E)\cup X_i(E')\) for all \(h=1,2, \ldots H\) and \(i\in N\).

Lemma 1

Let \(z_x\) denote the state of coordination on x. For all distinct pairs \((x,x') \in (X_A \cap X_B)^2\),

$$\begin{aligned} r(\mathcal {M}, z_x) \le r(z_{x'}, z_{x}). \end{aligned}$$

Proof

Let \(r(z_{x'}, z_{x})=a\). This means that for some \(k\in \{A,B\}\), after a errors by \(k'\) members, x is a best response for k members, and hence a weakly better response than a mutually impermissible action. That is:

$$\begin{aligned} d_k\le & {} \frac{a}{n}\theta _k + \delta _{kx}. \end{aligned}$$
(10)

Let \(\alpha _k\) be the minimum number of errors such that (10) holds:

$$\begin{aligned} \alpha _k = \bigg \lceil \frac{d_{k}-\delta _{kx}}{\theta _{k}}n\bigg \rceil \equiv \lceil D_{kx}n\rceil . \end{aligned}$$
(11)

We know \(r(z_{x'}, z_{x})=a\ge \min _{k\in \{A,B\}}\alpha _k\). By inspection of (10), \(\min _{k\in \{A,B\}}\alpha _k\) is the minimum cost of a direct path from any state in \(\mathcal {M}\) to \(z_x\). Hence \(r(\mathcal {M}, z_x)\le \min _{k\in \{A,B\}}\alpha _k\). This establishes the Lemma. \(\square \)

Lemma 2

Suppose that in state \(z, n_a\) members of group A and \(n_b\) members of group B use strategy x. Define \(z'\) as the state in which the same is true, with the remaining \(n-n_a\) members of group A and \(n-n_b\) members of group B using mutually impermissible actions.

Then \(r(z',z_x)\le r(z,z_x)\).

Proof

The argument in the proof of Lemma 1 establishes the result. \(\square \)

Lemma 3

If miscoordination \(\mathcal {M}\) is strictly risk dominant, then for n sufficiently large

$$\begin{aligned} r(z_{x}, \mathcal {M})<r(\mathcal {M}, z_x) \end{aligned}$$

for all \(x \in X_A \cap X_B\).

Proof

We claim that a direct path from \(\mathcal {M}\) to \(z_x\) has minimum cost among all paths from \(\mathcal {M}\) to \(z_x\). Consider an arbitrary indirect path from \(\mathcal {M}\) to \(z_x\). Since the path is indirect, it runs through a state z in which some mutually permissible action \(x'\ne x\) is played. The minimum cost of such a path is \(r(\mathcal {M},z) + r(z,z_x)\).

In state z, replace every action by a k member not equal to x with an action in \(X_{k}\backslash X_{k'}, k'\ne k\). Label this state \(z'\). There is a direct path from \(\mathcal {M}\) to \(z_x\) through state \(z'\). The minimum cost of such a path is \(r(\mathcal {M},z') + r(z',z_x)\). Clearly, \(r(\mathcal {M},z')<r(\mathcal {M},z)\). In addition, \(r(z', z_x) \le r(z,z_x)\) by Lemma 2. Hence the direct path has lower cost than the indirect path, establishing the claim.

By (11), the minimum cost of a direct path is \(\min _{k\in \{A,B\}}\lceil D_{kx} n\rceil \). Hence we need only find a path from \(z_{x}\) to \(\mathcal {M}\) with lower cost. Consider a direct path. Starting from \(z_x, k'\) members make a errors. For \(x''\in \widetilde{X}_{k}\backslash X_{k'}\) to be a best response for k members and to thereby reach a state in \(\mathcal {M}\) without further errors, a must satisfy

$$\begin{aligned} \bigg (1 - \frac{a}{n}\bigg )\theta _{k} + \delta _{kx}\le d_{k}. \end{aligned}$$

That is,

$$\begin{aligned} a\ge \bigg (1 - \frac{d_{k}-\delta _{kx}}{\theta _{k}}\bigg )n \equiv (1-D_{kx})n. \end{aligned}$$

Thus the minimum cost of such a transition is

$$\begin{aligned} \min _{k\in \{A,B\}}\big \lceil (1-D_{kx})n \big \rceil . \end{aligned}$$

For n sufficiently large, this is less than \(\min _{k\in \{A,B\}}\lceil D_{kx} n\rceil \) if and only if

$$\begin{aligned} \min _{k\in \{A,B\}}D_{kx}>&\min _{k\in \{A,B\}}\big (1 - D_{kx}\big )\nonumber \\ \min _{k\in \{A,B\}}D_{kx} + \max _{k\in \{A,B\}}D_{kx}>&1 \nonumber \\ D_{Ax} + D_{Bx}> & {} 1. \end{aligned}$$
(12)

The last inequality holds, because \(\mathcal {M}\) strictly risk dominates \(z_x\) [see (2)]. \(\square \)

The next result follows immediately from Lemmas 1 and 3:

Corollary 1

If miscoordination \(\mathcal {M}\) is strictly risk dominant, then for n sufficiently large

$$\begin{aligned} r(z_x,\mathcal {M}) < r(z_{x'}, z_{x}) \end{aligned}$$

for all distinct pairs \((x,x') \in (X_A \cap X_B)^2\).

Proof of Proposition 2 (i) Suppose that \(\mathcal {M}\) is strictly risk dominant. We shall show that for n sufficiently large, \(\mathcal {M}\) is the unique stochastically stable class using the spanning-tree method of Young (1993a).

The following tree-surgery argument is used. An x-tree is a tree consisting of a set of \(|X_A \cap X_B|\) directed edges, one emanating from each recurrent class (node) other than \(z_x\). Consider a tree, say rooted at 0 and denoted by \(T_0\), that has minimum resistance among x-trees. Through a series of operations on \(T_0\) we will produce an \(\mathcal {M}\)-tree with lower resistance than \(T_0\). Hence all trees with minimum resistance are \(\mathcal {M}\)-trees. By (Young 1993a, Theorem 2) then, \(\mathcal {M}\) is the unique stochastically stable class.

Again let \(T_0\) have minimum resistance among x-trees. By definition, there is a unique route from \(\mathcal {M}\) to \(z_{0}\) through \(T_0\) (see Fig. 1a for example). Denote the nodes on this route by \(z_{H}, z_{H-1}, \ldots z_1, z_{0}\). The edges composing this route are \((\mathcal {M}, z_{H}), (z_{H}, z_{H-1}), \ldots (z_{1}, z_{0})\).

Fig. 1
figure1

Beginning with the 0-tree in (a), steps 1–2 produce the \(\mathcal {M}\)-tree in (b)

  • Step 1. Replace edge \((\mathcal {M}, z_{H})\) with \((z_{H},\mathcal {M})\).

  • Step 2. Replace each edge \((z_{h}, z_{h-1})\) with \((z_{h-1}, \mathcal {M})\) for \(h=1,\ldots H\).

Steps 1–2 produce an \(\mathcal {M}\)-tree (see Fig. 1b for example). The difference in resistance between the original 0-tree and this \(\mathcal {M}\)-tree is:

$$\begin{aligned} \big [r(\mathcal {M}, z_{H}) - r(z_{H}, \mathcal {M})\big ] + \sum _{h=1}^{H}\big [r(z_{h}, z_{h-1}) - r(z_{h-1}, \mathcal {M})\big ].\end{aligned}$$
(13)

For n sufficiently large, the first term is positive by Lemma 3 and the second term, if it exists, is positive by Corollary 1. Hence the \(\mathcal {M}\)-tree produced by steps 1–2 has lower resistance than the original x-tree, \(T_0\). Since the original x-tree was chosen arbitrarily, all trees with minimum resistance are \(\mathcal {M}\)-trees, establishing the result.

(ii) Again a tree-surgery argument is employed. Consider an arbitrary \(\mathcal {M}\)-tree. There exists at least one edge \((z_{x}, \mathcal {M})\), for some \(x\in X_A \cap X_B\). Replace this edge with \((\mathcal {M}, z_{x})\). This transforms the original \(\mathcal {M}\)-tree into an x-tree. See Fig. 2 for an illustration.

Fig. 2
figure2

Beginning with the \(\mathcal {M}\)-tree in (a), replace edge \((z_{x}, \mathcal {M})\) with \((\mathcal {M}, z_{x})\) to produce the x-tree in (b)

The difference in resistance between the original \(\mathcal {M}\)-tree and this x-tree is:

$$\begin{aligned} r(z_{x}, \mathcal {M}) - r(\mathcal {M}, z_{x}). \end{aligned}$$

We claim that this is nonnegative when payoffs are simple and \(\mathcal {M}\) is not strictly risk dominant. Hence for every \(\mathcal {M}\)-tree there exists an x-tree with no greater resistance. By (Young 1993a, Theorem 2) then, \(\mathcal {M}\) is not the unique stochastically stable class.

Let us now establish the claim. In the proof of Lemma 3, we showed that \(r(\mathcal {M}, z_{x})=\min _{k\in \{A,B\}} \lceil D_{kx}n\rceil \) and the cost of a direct path from \(z_{x}\) to \(\mathcal {M}\) equals \(\min _{k\in \{A,B\}}\lceil (1-D_{kx})n \rceil \).

An indirect path from \(z_{x}\) to \(\mathcal {M}\) involves at least b erroneous plays of some mutually permissible action \(x'\ne x\) by \(k'\) members, so that \(x'\) is a best response for k members. Hence b satisfies:

$$\begin{aligned} \bigg (1-\frac{b}{n}\bigg )\theta _k + \delta _{kx}\le & {} \frac{b}{n}\theta _k + \delta _{kx'}\\ 2\frac{b}{n}\theta _k\ge & {} \theta _k + \delta _{kx} - \delta _{kx'}\\ b\ge & {} \frac{1}{2}n. \end{aligned}$$

The last line utilizes the fact that \(\delta _{kx}= \delta _{kx'}\) because payoffs are simple. Thus the cost of an indirect path from \(z_{x}\) to \(\mathcal {M}\) is at least \(\lceil \frac{1}{2}n\rceil \).

Hence when payoffs are simple, \(r(z_{x}, \mathcal {M})\ge r(\mathcal {M}, z_{x})\) if n is sufficiently large and

$$\begin{aligned} \min \bigg \{\min _{k\in \{A,B\}}(1-D_{kx}), \frac{1}{2}\bigg \} \ge \min _{k\in \{A,B\}} D_{kx}.\end{aligned}$$
(14)

We shall now show that (14) is satisfied. By hypothesis, \(\mathcal {M}\) is not strictly risk dominant. As payoffs are simple, this means that \(\mathcal {M}\) is risk dominated by all states of coordination. By (12) this is equivalent to

$$\begin{aligned} \min _{k\in \{A,B\}}D_{kx} \le \min _{k\in \{A,B\}}\big (1 - D_{kx}\big ), \end{aligned}$$
(15)

which implies

$$\begin{aligned} \min _{k\in \{A,B\}}D_{kx}\le & {} \max _{k\in \{A,B\}}\big (1 - D_{kx}\big ) \nonumber \\ \min _{k\in \{A,B\}}D_{kx}\le & {} 1 - \min _{k\in \{A,B\}}D_{kx}\nonumber \\ \min _{k\in \{A,B\}}D_{kx}\le & {} \tfrac{1}{2}. \end{aligned}$$
(16)

Together (15) and (16) imply that (14) holds. This establishes the claim and indeed part (ii) of the proposition.

Proof of Proposition 3 The result follows immediately from the fact that 1 / x is strictly positive, strictly decreasing and strictly convex.

Proposition 4 When \(\varepsilon =0\), there is no violation of cultural constraints. Hence the recurrent classes of the best response dynamic (\(\varepsilon =0\)) are still given by Proposition 1. These are the candidates for stochastic stability.

The only change in computation of resistances is to transitions from states of coordination to \(\mathcal {M}\). Hence Lemmas 1 and 2 still apply.

We shall now introduce two new lemmas and a corollary. Lemma 4 and Corollary 2 are analogs of Lemma 3 and Corollary 1, respectively, for the case of unconstrained errors. We shall then employ the tree-surgery arguments used in Proposition 2 to establish the result.

Lemma 4

If (9) is satisfied, then for n sufficiently large

$$\begin{aligned} r(z_{x}, \mathcal {M})<r(\mathcal {M}, z_x) \end{aligned}$$

for all \(x \in X_A \cap X_B\).

Proof

Recall that \(r(\mathcal {M},z_x) = \min _{k\in \{A,B\}}\lceil D_{kx} n\rceil \).

Let us now compute \(r(z_x,\mathcal {M})\). On the minimum cost path from \(z_x\) to \(\mathcal {M}\) one of two things must occur. Starting from \(z_x\), through a series of errors, either (i) a mutually impermissible action becomes a best response, in which case the process transits to \(\mathcal {M}\) without any further errors, or (ii) an action \(x'\in X_A \cap X_B\) becomes a best response.

The minimum cost path conforming to case (i) is a direct path as follows. As errors are unconstrained, a transition from \(z_x\) to \(\mathcal {M}\) can occur with a erroneous plays of an action in \(\widetilde{X}_k\setminus X_{k'}\) by \(k'\) members, where a satisfies

$$\begin{aligned} \bigg (1 - \frac{a}{n}\bigg )\theta _k + \delta _{kx} \le \frac{a}{n}\theta _k + d_{k}. \end{aligned}$$
(17)

The value of a that equates (17) is \((1-D_{kx})\frac{n}{2}\). Hence the cost of such a path from \(z_x\) to \(\mathcal {M}\) is

$$\begin{aligned} \min _{k\in \{A,B\}} \bigg \lceil (1-D_{kx})\frac{n}{2}\bigg \rceil = \min _{k\in \{A,B\}} \bigg \lceil \bigg (1-\frac{d_{k}-\delta _{kx}}{\theta _k}\bigg )\frac{n}{2}\bigg \rceil . \end{aligned}$$
(18)

The minimum cost path conforming to case (ii) involves at least b erroneous plays of some mutually permissible action \(x'\ne x\) by \(k'\) members, so that \(x'\) is a best response for k members. Hence b satisfies

$$\begin{aligned} \bigg (1 - \frac{b}{n}\bigg )\theta _k + \delta _{kx} \le \frac{b}{n}\theta _k + \delta _{kx'}. \end{aligned}$$

Thus the cost of such a path from \(z_{x}\) to \(\mathcal {M}\) is at least

$$\begin{aligned} \min _{k\in \{A,B\}} \bigg \lceil \bigg (1-\frac{\delta _{kx'}-\delta _{kx}}{\theta _k}\bigg )\frac{n}{2}\bigg \rceil . \end{aligned}$$
(19)

This is greater than (18) because \(d_{k}>\delta _{kx'}\) by misalignment. Hence the minimum cost path from \(z_x\) to \(\mathcal {M}\) is direct and \(r(z_x,\mathcal {M})\) equals (18).

Therefore, when errors are unconstrained and n is sufficiently large the following statements are equivalent:

$$\begin{aligned} r(\mathcal {M},z_x)>&r(z_x, \mathcal {M}),\nonumber \\ \min _{k\in \{A,B\}}D_{kx}>&\min _{k\in \{A,B\}}\tfrac{1}{2}\big (1 - D_{kx}\big ),\nonumber \\ 2\min _{k\in \{A,B\}}D_{kx} + \max _{k\in \{A,B\}}D_{kx}> & {} 1. \end{aligned}$$
(20)

The last line holds by hypothesis (9). \(\square \)

The next result follows immediately from Lemmas 1 and 4.

Corollary 2

If (9) is satisfied, then for n sufficiently large

$$\begin{aligned} r(z_x,\mathcal {M}) < r(z_{x'}, z_{x}) \end{aligned}$$

for all distinct pairs \((x,x') \in (X_A \cap X_B)^2\).

Lemma 5

\(r(z_x, \mathcal {M}) \le r(z_x,z_{x'})\) for all distinct \((x,x')\in (X_A\cap X_B)^2\).

Proof

Recall that \(r(z_x, \mathcal {M})\) equals (18).

Now consider \(r(z_x,z_{x'})\). On the minimum cost path from \(z_x\) to \(z_{x'}\) one of two things must occur. Starting from \(z_x\), through a series of errors, either (i) a mutually impermissible action becomes a best response, in which case \(r(z_x,z_{x'})\ge r(z_x, \mathcal {M})\), or (ii) an action \(x''\in X_A \cap X_B\) (possibly \(x'\)) becomes a best response. The path with minimum cost that conforms to the case (ii) involves \(\beta \) erroneous plays of \(x''\) by \(k'\) members, where \(\beta \) equals (19).

We established in the proof of Lemma 4 that (19) is greater than (18). This establishes the Lemma. \(\square \)

Proof of Proposition 4 To prove that (9) is sufficient for \(\mathcal {M}\) to be stochastically stable we use the same tree-surgery argument as in Proposition 2(i). Starting with an arbitrary x-tree, we transform it through steps 1–2 into an \(\mathcal {M}\)-tree. Again the difference in resistance between the original x-tree and the resultant \(\mathcal {M}\)-tree is given by (13). If (9) holds, then the first term is positive by Lemma 4 and the second term, if it exists, is positive by Corollary 2. Hence (9) is sufficient.

To establish that (9) is also necessary, note that Lemma 5 implies that the minimum cost \(\mathcal {M}\)-tree is composed entirely of direct links, i.e., edges \((z_x, \mathcal {M})\) for each \(x\in X_A\cap X_B\).

Suppose that (9) were violated, so that \(2\min _{k\in \{A,B\}}D_{kx} + \max _{k\in \{A,B\}}D_{kx} \le 1\) for some \(x\in X_A\cap X_B\). Recall that the minimum cost \(\mathcal {M}\)-tree contains edge \((z_x, \mathcal {M})\). Replace this edge with \((\mathcal {M},z_x)\). This operation transforms the minimum cost \(\mathcal {M}\)-tree into an x-tree. The difference in resistance between the original \(\mathcal {M}\)-tree and the resultant x-tree is \(r(z_x, \mathcal {M})-r(\mathcal {M},z_x)\) which is nonnegative as (9) is violated [see (20)]. Hence \(\mathcal {M}\) is not the unique stochastically stable class. \(\square \)

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Carvalho, J. Coordination and culture. Econ Theory 64, 449–475 (2017). https://doi.org/10.1007/s00199-016-0990-3

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Keywords

  • Culture
  • Conflict
  • Coordination failure
  • Stochastic stability

JEL Classification

  • C72
  • C73
  • Z1