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Group Identities in Conflicts

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Abstract

If the members of a group identify with their group, free-riding behavior within the group is reduced. This seems beneficial at first sight. However, in contests between groups, identification escalates conflicts, increasing rent dissipation and possibly generating welfare losses. Generally, in an inter-group conflict the adoption or non-adoption of a group identity is endogenous. We show that, if groups are similar in size and conflict technology, all groups will adopt a group identity, reducing welfare for all. If groups are unequal, the stronger one will develop a group identity, which goes at the expense of the weaker group. Out-group hostility favors asymmetric identities. Applications include team spirit in war and sports, national identities or (seemingly) dysfunctional behavior of social groups.

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Notes

  1. For surveys on contests between groups see Garfinkel and Skaperdas (2007, Section 7) or Konrad (2009, Chapters 5.5 and 7).

  2. Predictions for asymmetric group contests have—to our knowledge—not yet been tested in the laboratory.

  3. Weisel and Zultan (2016) argue that the framing of the conflict as a threat against individuals or against their group plays an important role, too.

  4. Alternatively the rent could be a group-specific public good. Qualitatively, our results extend to any other sharing rule that does not adjudge the complete marginal return on the investment to individuals.

  5. This specification is a tractable case of the approach in Shayo (2007); what is essential is that decision utilities with a group identity are strictly monotonic, welfarist function that aligns individual incentives and group welfare.

  6. This weakest-link assumption seems to hold in war-like situations (Keegan 1976). It buys us a simple game since we need not consider all possible vectors of individual identities that might emerge within a group. We may lose many asymmetric equilibria, which are, however, of secondary interest. In a richer model the “unanimity” assumption (3) could be replaced by a trigger function, conveying that a group identity prevails once a certain threshold of support is reached. Likewise, binary identities could be replaced by continuous variables that measure various degrees of identification (Abdelal et al. 2009). As long as the identity equilibrium is unique this would, however, not qualitatively affect our results.

  7. The terminology of choice and decision that is standard in economics may be misleading when applied to the processes that determine group identities. The terminology suggests a conscious and deliberate process of decision making by economic agents—while nothing in the leading paradigm of “mindless economics” (Dietrich and List 2016; Hausman 2013; Hands 2009) requires consciousness to understand the links between stimuli and responses. In particular, mental phenomena like the identification with a group may fully be understood as a mainly emotional, subconscious mechanisms that nevertheless have behavioral (and perceptional) consequences. However, the identity that is actually triggered is not arbitrary but depends on either physical (like skin color) or cultural markers (like religion) that are shaped in a long-term process of memetic evolution. See Appendix A.4 for further details.

  8. In addition, trembling-hand perfection narrows down the set of equilibria in simultaneous games like ours. Alternative concepts like backward or forward induction, subgame perfection, perfect Bayesian equilibria etc. only allow for equilibria selection in sequential games.

  9. Explicit expressions for \({\underline{\theta }}(z,N_A,N_D)\) and \({\bar{\theta }}(z,N_A,N_D)\) are provided in Appendix A.2. Their analysis does not yield any exciting insights, however.

  10. The literature that focuses on strong identities in small groups must have a different mechanism that explains the phenomenon. Given that individuals are homogenous in our model, we isolate the pure effect of group size (corrected by \(\theta \) and z), and this effect is unambiguous: the potential to internalize a comparatively larger within-group externality is decisive. Hence, if this logic is correct, the reason for why small groups sometimes end up with stronger identities may point towards the importance of within-group heterogeneity (which is likely to be smaller in smaller groups).

  11. See Kolmar and Rommeswinkel (2018) for a general analysis of standard group-contests that allows for heterogeneity within groups.

  12. If intrinsic utility gains from group identification mattered, any welfarist assessment that relies on the idea of normative individualism runs into problems (Fang and Loury 2005). In addition, the model could easily become meaningless in the sense that sufficiently large “empathy rents” from identifying with the rest of the group may easily outweigh any rise in material payoffs. Bierbrauer and Netzer (2016) discuss this issue in the context of mechanism design.

  13. Note a problem of causality here: many empirical studies show that social identities facilitate commitment, implying that they themselves can be interpreted as a commitment mechanism (Burke and Reitzes 1991). At the same time, commitment is necessary to sustain collective identities (Hunt and Benford 2004).

  14. We thank one of the referees for bringing this up.

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Appendix

Appendix

1.1 Equilibria of the Second-Stage Subgame

Suppressing constant parameters in notation, denote by \(V_A(\alpha , \delta )\) and \(V_D(\alpha , \delta )\) the subgame-equilibrium levels of \(u_A^i\) and \(u_D^i\) for members of groups A and D, respectively (individual indexes can be dropped due to the symmetry within groups). Formally, \(V_A\) and \(V_D\) are the Nash equilibrium values of \(u_A^i\) and \(u_D^i\) when individual group members each solve the problems \(\max _{a_i} u_A^i(a_i,a_{-i},d, \alpha , \delta )\) and \(\max _{_i} u_D^i(a, d_i,d_{-i},\alpha , \delta )\), respectively.

Since \((\alpha , \delta ) \in \{0,1\}^2\), four cases can arise.

Case 1 Both groups have an individualistic identity. If \((\alpha , \delta )=(0,0)\), a representative member of group A, D solves the following problem:

$$\begin{aligned} \max _{a_i} u_A^i(a_i,a_{-i},d,0, 0), \quad \text{ and } \quad \max _{_i} u_D^i(a, d_i,d_{-i},0, 0). \end{aligned}$$

The Nash equilibrium in this subgame is given by

$$\begin{aligned} a_i( 0, 0) = \frac{N_D R \theta }{N_A(N_A + N_D \theta )^2}, \quad d_i( 0, 0) = \frac{N_A R \theta }{N_D(N_A + N_D \theta )^2}. \end{aligned}$$
(5)

Hence, investments are decreasing in the size of one’s own group: in a larger group, effect of an individual’s contribution on the outcome gets smaller and incentives to free-ride get stronger.

Without group identities, individual payoffs \(u_j^i\) coincide with material payoffs, \(\pi _i^k\):

$$\begin{aligned} V_A(0,0) = \frac{N_D R \theta (N_A + N_D \theta - 1)}{N_A(N_A + N_D \theta )^2}, \quad V_D(0,0) = \frac{N_A R(N_A + (N_D - 1)\theta )}{N_D(N_A + N_D \theta )^2}. \end{aligned}$$
(6)

Case 2 Only group D has a group identity. With \((\alpha , \delta )=(0,1)\), representative members of group A and D solve the following problems:

$$\begin{aligned} \max _{a_i} u_A(a_i,a_{-i},d,0,1), \quad \text{ and } \quad \max _{d_i} u_D(a, d_i,d_{-i},0,1). \end{aligned}$$

The Nash equilibrium of this subgame is given by

$$\begin{aligned} a_i(0,1) = \frac{R \theta (1+z)}{N_A ((1+z) N_A + \theta )^2},\quad d_i(0,1) = \frac{R N_A \theta (1+z)^2}{N_D ((1+z) N_A + \theta )^2}. \end{aligned}$$
(7)

The associated per-capita material welfare levels are

$$\begin{aligned} V_A(0,1)= & {} \frac{R \theta ((N_A-1)(1+z)+ \theta )}{N_A ((1+z) N_A + \theta )^2},\\ V_D(0,1)= & {} \frac{R N_A (1+z) (N_A(1+z)- \theta z)}{N_D ((1+z) N_A + \theta )^2}.\nonumber \end{aligned}$$
(8)

Case 3 Only group A has a group identity. The case \((\alpha , \delta )=(1,0)\) is essentially a permutation of case 2. Per-capita material welfare levels amount to:

$$\begin{aligned} V_A(1,0)= & {} \frac{R N_D \theta (1+z) (N_D \theta (1+z)-z)}{N_A (N_D \theta (1+z)+1)^2},\nonumber \\ V_D(1,0)= & {} \frac{R(\theta (N_D-1) (1+z)+1)}{N_D (N_D \theta (1+z)+1)^2}. \end{aligned}$$
(9)

Case 4 Both groups have a group identity. With \((\alpha , \delta )=(1,1)\) the individual optimization problems in groups A and D are:

$$\begin{aligned} \max _{a_i} u_A(a_i,a_{-i},d,1, 1), \quad \text{ and } \quad \max _{d_i} u_D(a, d_i,d_{-i},1,1). \end{aligned}$$

In the Nash equilibrium of this subgame,

$$\begin{aligned} a_i(1,1) = \frac{R \theta (1+z)}{N_A (1+\theta )^2},\quad d_i(1,1) = \frac{R \theta (1+z)}{N_D (1+\theta )^2}, \end{aligned}$$
(10)

individuals capture rents of

$$\begin{aligned} V_A(1,1)= & {} \frac{R \theta (\theta -z)}{N_A (1+\theta )^2}, \nonumber \\ V_D(1,1)= & {} \frac{R (1-\theta z)}{N_D (1+\theta )^2}. \end{aligned}$$
(11)

Comparisons While a full comparison of the various Nash equilibria is only marginally relevant, some aspects deserve mention. Differences in group sizes (\(N_A \ne N_D\)) and productivities (\(\theta \ne 1\)) shape the intensity of conflicts in a complex way. Comparing (10) and (5) for groups of equal size (\(N_A=N_D\)) shows that group identities lead to a higher conflict intensity. Comparing, for identical group sizes, contest efforts \(a_i\) and \(d_i\) in the asymmetric cases 2 and 3, individual efforts are higher in groups with a group identity, compared to individualistic groups. Sharper hostility (a higher level of z) leads to more intense conflicts.

1.2 Proof of Proposition 1

Proposition 1 is a corollary of Proposition 3 below. Its proof requires some preliminaries.

Let \(\Gamma = \{N_A, N_D, \{\alpha _i\}_{i=1,\ldots ,N_A},\{\delta _j\}_{j=1,\ldots ,N_D}, \alpha (.), \delta (.), V_A(.), V_D(.)\}\) be the strategic form of the first-stage game. For \(i=1,\ldots ,N_A\), denote by \(\alpha _i^M\) a mixed strategy for \(\alpha _i\) (i.e., a probability that \(\alpha _i=1\) is played) and, likewise, by \(\delta _j^M\) a mixed strategy on \(\delta _j\) (with \(j=1,\ldots ,N_D\)). The corresponding game in mixed strategies is defined by \(\Gamma ^M = \{N_A, N_D, \{\alpha _i^M\}_{i=1,\ldots ,N_A},\{\delta _j^M\}_{j=1,\ldots ,N_D}, \alpha (.), \delta (.), E[V_A(.)], E[V_D(.)]\}\), where we have assumed that individuals maximize their expected material payoff and E[.] denotes the expectations operator. A perturbed game \(\Gamma ^{P}\) is a game \(\Gamma ^{M}\) that allows only for totally mixed strategies \(\alpha _i^M \in (0,1)\), \(\delta _j^M \in (0,1)\).

Definition

A strategy profile \(\{\alpha _i^*\}_{i=1,\ldots ,N_A},\{\delta _j^*\}_{j=1,\ldots ,N_D}\) in \(\Gamma \) is a trembling-hand perfect Nash equilibrium if there is a sequence of perturbed games \(\Gamma ^{P}\), converging to \(\Gamma \), for which the sequence of Nash equilibria \(\{\alpha _i^{M*}\}_{i=1,\ldots ,N_A},\{\delta _j^{M*}\}_{j=1,\ldots ,N_D}\) converges to \(\{\alpha _i^*\}_{i=1,\ldots ,N_A},\{\delta _j^*\}_{j=1,\ldots ,N_D}\).

Proposition 3

There exist threshold values \({\hat{z}} \in (0,1)\) , \({\underline{\theta }}(z,N_A,N_D)\) and \({\bar{\theta }}(z,N_A,N_D)\) with \(0< {\underline{\theta }}(z,N_A,N_D)< {\bar{\theta }}(z,N_A,N_D)\) such that the following holds:

  • If \( z < {\hat{z}}\) and

    1. 1.

      if \(\theta < {\underline{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 0\) for \(i=1,\ldots ,N_A\) and \(\delta _j^* = 1\) for \(j= 1,\ldots ,N_D\) ;

    2. 2.

      if \({\underline{\theta }}(z,N_A,N_D)< \theta < {\bar{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 1\) for \(i=1,\ldots ,N_A\) , and \(\delta _j^* = 1\) for \(j= 1,\ldots ,N_D\) ;

    3. 3.

      if \(\theta > {\bar{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 1\) for \(i=1,\ldots ,N_A\) , and \(\delta _j^* = 0\) for \(j= 1,\ldots ,N_D\) ;

    4. 4.

      if \(\theta = {\underline{\theta }}(z,N_A,N_D)\) or \(\theta = {\bar{\theta }}(z,N_A,N_D)\) , there exist two trembling-hand perfect Nash equilibria, \(\alpha _i^* = 0, \delta _j^* = 1\) and \(\alpha _i^* = 1, \delta _j^* = 1\) , and \(\alpha _i^* = 1, \delta _j^* = 0\) and \(\alpha _i^* = 1, \delta _j^* = 1\) , respectively.

  • If \( z > {\hat{z}}\) and

    1. 1.

      if \(\theta < {\underline{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 0\) for \(i=1,\ldots ,N_A\) , and \(\delta _j^* = 1\) for \(j= 1,\ldots ,N_D\) ;

    2. 2.

      if \({\underline{\theta }}(z,N_A,N_D) \le \theta \le {\bar{\theta }}(z,N_A,N_D)\) , there exist two trembling-hand perfect Nash equilibria, \(\alpha _i^* = 0, \delta _j^* = 1\) , and \(\alpha _i^* = 1, \delta _j^* = 0\) (with \(i=1,\ldots ,N_A\) and \(j= 1,\ldots ,N_D\) );

    3. 3.

      if \(\theta > {\bar{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 1\) for \(i=1,\ldots ,N_A\) , and \(\delta _j^* = 0\) for \(j= 1,\ldots ,N_D\) .

  • If \( z = {\hat{z}}\) and

    1. 1.

      if \(\theta < {\underline{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 0\) for \(i=1,\ldots ,N_A\) , and \(\delta _j^* = 1\) for \(j= 1,\ldots ,N_D\) ;

    2. 2.

      if \({\underline{\theta }}(z,N_A,N_D) \le \theta \le {\bar{\theta }}(z,N_A,N_D)\) , there exist three trembling-hand perfect Nash equilibria, \(\alpha _i^* = 0, \delta _j^* = 1, \alpha _i^* = 1, \delta _j^* = 0\) , and \(\alpha _i^* = 1,\delta _j^* = 1\) (with \(i=1,\ldots ,N_A\) and \(j= 1,\ldots ,N_D\) );

    3. 3.

      if \(\theta > {\bar{\theta }}(z,N_A,N_D)\) , the unique trembling-hand perfect Nash equilibrium is \(\alpha _i^* = 1\) for \(i=1,\ldots ,N_A\) , and \(\delta _j^* = 0\) for \(j= 1,\ldots ,N_D\) .

Proof of Proposition 3

The payoffs for given identities \((\alpha , \delta )\) are given in the following matrix:

$$\begin{aligned} \begin{array}{c|c|c} &{} \delta = 1 &{} \delta = 0\\ \hline \alpha = 1 &{} V_A(1,1),V_D(1,1) &{} V_A(1,0),V_D(1,0) \\ \hline \alpha = 0 &{} V_A(0,1),V_D(0,1) &{} V_A(0,0), V_D(0,0) \end{array} \end{aligned}$$

GroupD (i) Assume that individuals of group A independently play \(\alpha _i = 0\) with probability \(1-\epsilon _A, \epsilon _A > 0\). In that case, members of the group have an individualistic identity with probability \(1 - \epsilon _A^{N_A}\). Assume in addition that individuals of group D independently play \(\delta _j = 0\) with probability \(1-\epsilon _D, \epsilon _D > 0\). For \(\epsilon _A \rightarrow 0\), the utility differential that results from the creation of a group identity for a member of group D is

$$\begin{aligned} \Delta _D(\alpha =0)= & {} V_D(0, 1) -V_D(0, 0) \\= & {} \left( \frac{(z+1) (z N_A+N_A-\theta z)}{(z N_A+N_A+\theta )^2}-\frac{N_A+(N_D-1) \theta }{(N_A+N_D \theta )^2}\right) \frac{N_A R}{N_D}. \end{aligned}$$

This is non-negative if and only if

$$\begin{aligned} \theta \le \theta _D^1 := \frac{N_A (N_D+(N_D-2) z-1)+\sqrt{N_A^2 \left( N_D^2 (z+1)^2-2 N_D (z+1)+4 z (z+2)+5\right) }}{2 N_D z+2} > 0. \end{aligned}$$

Individual k of group D is only decisive in influencing the group identity if all other members of group D vote \(\delta _j = 1\), with happens with probability \(\epsilon _D^{N_D-1} > 0\) for all \(\epsilon _D > 0\). Hence, an individual of group D is better off adopting a group identity.

(ii) Assume that individuals of group A independently play \(\alpha _i = 1\) with probability \(1-\epsilon _A, \epsilon _A > 0\). In that case, members of the group have a group identity with probability \(1 - \epsilon _A^{N_A}\). Assume in addition that individuals of group D independently play \(\delta _j = 0\) with probability \(1-\epsilon _D, \epsilon _D > 0\). For \(\epsilon _A \rightarrow 0\), the utility differential that results from the creation of a group identity for a member of group D is

$$\begin{aligned} \Delta _D(\alpha =1)= & {} V_D(1, 1) -V_D(1, 0) \\= & {} \left( \frac{1-\theta z}{(\theta +1)^2}-\frac{(N_D-1) \theta (z+1)+1}{(N_D \theta (z+1)+1)^2}\right) \frac{R}{N_D}. \end{aligned}$$

This is non-negative if and only if

$$\begin{aligned} \theta \le \theta _D^2 := \frac{2}{2 z-N_D (z+1)+\sqrt{N_D^2 (z+1)^2-2 N_D (z+1)+4 z (z+2)+5}+1} > 0. \end{aligned}$$

Individual k of group D is only decisive in influencing the group identity if all other members of group D choose \(\delta _j = 1\), with happens with probability \(\epsilon _D^{N_D-1} > 0\) for all \(\epsilon _D > 0\). Hence, an individual of group D is better off adopting a group identity.

GroupA (iii) Assume that individuals of group D independently play \(\delta _j = 0\) with probability \(1-\epsilon _D, \epsilon _D > 0\). In that case, members of the group have an individualistic identity with probability \(1 - \epsilon _D^{N_D}\). Assume in addition that individuals of group A independently play \(\alpha _i = 0\) with probability \(1-\epsilon _A\), with \(\epsilon _A > 0\). For \(\epsilon _D \rightarrow 0\), the utility differential that results from the creation of a group identity for a member of group A is

$$\begin{aligned} \Delta _A(\delta =0)& = \,V_A(1, 0) -V_A(0, 0) \\&= \left( \frac{(z+1) (N_D \theta (z+1)-z)}{(N_D \theta (z+1)+1)^2}-\frac{N_A+N_D \theta -1}{(N_A+N_D \theta )^2}\right) \frac{N_D \theta R}{N_A}, \end{aligned}$$

which is non-negative if and only if

$$\begin{aligned} \theta\ge & {} \theta _A^1 \\:= & {} \frac{\sqrt{N_D^2 \left( N_A^2 (z+1)^2-2 N_A (z+1)+4 z (z+2)+5\right) }-N_D (N_A+(N_A-2) z-1)}{2 N_D^2(z+1)}, \end{aligned}$$

where \(\theta _A^1\) is positive. Individual k of group A is only decisive in influencing the group identity if all other members of group A set \(\alpha _i = 1\), with happens with probability \(\epsilon _A^{N_A-1} > 0\) for all \(\epsilon _A > 0\). Hence, an individual of group D is better off adopting a group identity.

(iv) Assume that individuals of group D independently play \(\delta _j = 1\) with probability \(1-\epsilon _D, \epsilon _D > 0\). In that case, members of the group have a group identity with probability \(1 - \epsilon _D^{N_D}\). Assume in addition that individuals of group A independently play \(\alpha _i = 0\) with probability \(1-\epsilon _A, \epsilon _A > 0\). For \(\epsilon _D \rightarrow 0\), the utility differential that results from a group identity for a member of group A is

$$\begin{aligned} \Delta _A(\delta =1)= & {} V_A(1, 1) -V_A(0, 1) \\= & {} \left( \frac{\theta -z}{(\theta +1)^2}-\frac{N_A+\theta +(N_A-1) z-1}{(z N_A+N_A+\theta )^2}\right) \frac{\theta R}{N_A} , \end{aligned}$$

which is non-negative if and only if

$$\begin{aligned} \theta \le \theta _A^2 := \frac{1}{2} \left( 2 z-N_A (z+1)+\sqrt{N_A^2 (z+1)^2-2 N_A (z+1)+4 z (z+2)+5}+1\right) , \end{aligned}$$

where \(\theta _A^2\) is positive. Member k of group A is only decisive in influencing the group identity if all other members of group A choose \(\alpha _i = 1\), which happens with probability \(\epsilon _A^{N_A-1} > 0\) for all \(\epsilon _A > 0\). Hence, a member of group A is better off adopting a group identity.

(v) It is straightforward to show that \(\theta _A^1 <\theta _A^2\) and \(\theta _D^2 <\theta _D^1\). Depending on z, we get the following inequalities:

  • If \(z < {\hat{z}}\), it follows that \(\theta _A^1< \theta _A^2< \theta _D^2 < \theta _D^1\).

  • If \(z > {\hat{z}}\), it follows that \(\theta _A^1< \theta _D^2< \theta _A^2 < \theta _D^1\).

  • If \(z = {\hat{z}}\), it follows that \(\theta _A^1< \theta _D^2 = \theta _A^2 < \theta _D^1\).

Here, \({\hat{z}}\) is implicitly defined by \(\Psi ({\hat{z}}) := \theta _D^2({\hat{z}}) - \theta _A^2 ({\hat{z}}) = 0\).

Let \(z < {\hat{z}}\). It follows that \(\alpha _i = 0\), \(\delta _j = 1\) (for all \(i=1,\ldots ,N_A\), \(j=1,\ldots ,N_D\)) is the trembling-hand perfect equilibrium for \(\theta \in [0,\theta _A^2)\). For \(\theta \in (\theta _A^2, \theta _D^2)\) it follows that \(\alpha _i = 1\), \(\delta _j = 1\) for all ij is the trembling-hand perfect equilibrium. For \(\theta \in (\theta _D^2, \infty )\) the equilibrium is at \(\alpha _i = 1\), \(\delta _j = 0\) for all ij. Finally, for the boundary cases \(\theta = \theta _A^2\) and \(\theta = \theta _D^2\) the equilibria from both connecting intervals remain equilibria. Putting \({\underline{\theta }}= \theta _D^2\) and \({\bar{\theta }}= \theta _A^2\), the claim follows.

Let \(z > {\hat{z}}\). With the above utility differentials it follows that \(\alpha _i = 0\), \(\delta _j = 1\) for \(i=1,\ldots ,N_A\), \(j=1,\ldots ,N_D\) is the trembling-hand perfect equilibrium for \(\theta \in [0,\theta _D^2)\). For \(\theta \in [\theta _D^2, \theta _A^2]\) the equilibrium is \(\alpha _i = 0\), \(\delta _j = 1\) as well as \(\alpha _i = 1\), \(\delta _j = 0\) for all ij are trembling-hand perfect equilibria. Finally, for \(\theta \in (\theta _A^2, \infty )\), \(\alpha _i = 1\), \(\delta _j = 0\) (for all ij) is the trembling-hand perfect equilibrium. Putting \({\underline{\theta }}= \theta _A^2\) and \({\bar{\theta }}= \theta _D^2\), the claim follows.

If \(z = {\hat{z}}\) one gets \(\theta _A^2 = \theta _D^2\). In this case, \(\alpha _i = 0\), \(\delta _j = 1\) for all ij is the trembling-hand perfect equilibrium for \(\theta \in [0,\theta _D^2)\); \(\alpha _i = 0\), \(\delta _j = 1\) is the trembling-hand perfect equilibrium for \(\theta \in (\theta _D^2, \infty )\). There are three trembling-hand perfect equilibria at \(\theta =\theta _D^2\): \(\alpha _i = 0\), \(\delta _j = 1\); \(\alpha _i = 1\), \(\delta _j = 0\); and \(\alpha _i = 1\), \(\delta _j = 1\). Putting \({\underline{\theta }}= {\bar{\theta }}= \theta _A^2 = \theta _D^2\), the claim follows. \(\square \)

1.3 Proof of Proposition 2

  1. 1.

    Assume there exists an equilibrium with \(\alpha = 0, \delta = 1\). Then members of group D must be better off by revealed preference. Members of group A are not worse off if and only if \(V_A(0,1) \ge V_A(0,0)\), which is equivalent to

    $$\begin{aligned}&\frac{N_A+\theta +(N_A-1) z-1}{(z N_A+N_A+\theta )^2}-\frac{N_D (N_A+N_D \theta -1)}{(N_A+N_D \theta )^2} \ge 0 \nonumber \\&\quad \Leftrightarrow -N_A (z N_A+N_A-1) \left( (\theta +1) (z+1) N_A^2+\left( \theta ^2+\theta -z-1\right) N_A+\theta ^2\right) \ge 0\nonumber \\&\quad \Leftrightarrow (N_A-1)(z+1)+\theta ((1+z)N_A + (1+\theta )) \le 0, \end{aligned}$$
    (12)

    which, however, contradicts the assumption that \(N_A, N_D \ge 2\).

    Next assume there exists an equilibrium with \((\alpha , \delta )=(1,0)\). Individuals of group A must be better off than at (0, 0) by revealed preferences. Individuals of group D are not worse off if and only if \(V_D(1,0) \ge V_D(0,0)\), which is equivalent to

    $$\begin{aligned} \frac{N_A+\theta +(N_A-1) z-1}{(z N_A+N_A+\theta )^2}-\frac{N_D (N_A+N_D \theta -1)}{(N_A+N_D \theta )^2} \ge 0 \end{aligned}$$

    or:

    $$\begin{aligned} -\theta (z N_A+N_A-1) \left[ \theta ((N_D-1) \theta (z+1)+1) N_D^2+N_A \left( \theta (z+1) N_A^2+N_D+1\right) \right] \ge 0. \end{aligned}$$

    This again contradicts the assumption that \(N_A \ge 2\) since the square-bracketed expression is positive.

  2. 2.

    If \((\alpha , \delta )=(1,1)\), group A is better off than in the individualistic case iff \(V_A(1,1) \ge V_A(0,0)\) and group D is better off iff \(V_D(1,1) \ge V_D(0,0)\). These conditions are equivalent to

    $$\begin{aligned}&A:&\frac{\theta -z}{(\theta +1)^2}-\frac{N_D (N_A+N_D \theta -1)}{(N_A+N_D \theta )^2}\ge 0,\nonumber \\&D:&\frac{1-\theta z}{(\theta +1)^2}-\frac{N_A (N_A+(N_D-1) \theta )}{(N_A+N_D \theta )^2} \ge 0. \end{aligned}$$
    (13)
    • If \(N_A=N_D=N\), these conditions both require

      $$\begin{aligned} \frac{1-N (1+z)}{N (\theta +1)^2} \ge 0, \end{aligned}$$

      which cannot hold for \(N>1\). Hence, both groups are worse off with group identities if they have similar sizes.

    • For general population structures, conditions (13) run into opposite directions for groups A and D. Group A is better off and group D is worse off if \(N_A\) is sufficiently larger than \(N_D\) and \(\theta \) is relatively large. The converse holds for large \(N_D\) and small \(\theta \).

    • For general population structures, we jointly analyzed (13) using the mathematica software: running “\( Reduce[V_A(1,1) -V_A(0,0) \ge 0 \& \& V_D(1,1) -V_D(0,0) \ge \& \& Na \ge 2 \& \& Nd \ge = 2 \& \& z \ge 0 \& \& theta \ge 0, theta]\)” generates “False” as the output. Hence, group identities can never lead to a Pareto improvement over the no-identity case. q.e.d.

1.4 Identities as Emotions: A Survey

Damasio (2010), Damasio et al. (2013) and LeDoux (2002, 2015) distinguish between emotions, feelings, and affects. Emotions are changes in both, body and brain states in response to external or internal stimuli. They are part of a homeostatic regulation system that helps the organism to survive and reproduce. By this definition, emotions as dynamic response mechanisms are not conscious. Feelings, however, are. Feelings are the equivalent of emotions that pass the threshold of consciousness. The term affect encompasses both, emotions and feelings.

Research from neuroscience and psychology (see, e.g., Öhmann and Wiens 2003) suggests that the mechanism that triggers behavior as a response to stimuli is a complex multi-layered process of emotional as well as cognitive reactions, where the heavy lifting is done by unconscious emotional mechanisms (LeDoux 2002, 2015; Damasio 2010). The stimuli and experiences that reach the level of consciousness always carry an “affective load”, a complex feeling that colors perceptions and influences behavior. This implies that the conscious, narrative interpretation of the exteroceptive (group conflict) and interoceptive (emotional state) gives rise to a perceived identity (Seth 2013).

Given the biochemical roots of parochial altruism versus individualism, group identities may have their roots in the emotional mechanisms of human brains that get activated by certain environmental stimuli. This conjecture is supported by De Dreu and Nauta (2009); De Dreu (2012) who identify oxytocine as an important hormone regulating group behavior, and by Tausch et al. (2011) who show that an emotional pathway associated with feelings like anger or contempt is correlated with triggering collective action in group conflicts. Reimers and Diekhof (2015) show that high testosterone levels are associated with increased in-group cooperation during intergroup competition. De Dreu et al. (2015) found that in an intergroup prisoner’s dilemma self-sacrificial decisions to contribute were made faster than decisions not to contribute. In addition, lowering impulse control (or reducing deliberation) increased behavior that benefited in-group members while harming out-group members. In line with SIT, Freeman and Ambady (2011) show that group categorization is quick and effortless and can shift dynamically as a function of bottom-up sensory information and top-down social goals and motivations. All these findings support the view that behaviorally relevant (as opposed to latent) group identities are automatically and quickly triggered via emotional pathways.

Our proposed equilibrium mechanism is supported by the literature on group emotional contagion. In an empirical experiment, Barsade (2002) could show that mood can be quickly transferred in a group, and that there is a significant influence of emotional contagion (as an implicit, automatic, emotional mechanism) on individual-level attitudes and group processes, leading to increased in-group cooperation. Given that these effects can be triggered in an experimental setting and in line with the minimal group paradigm, they are quick and do not rely on long-term cultural processes.

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Kolmar, M., Wagener, A. Group Identities in Conflicts. Homo Oecon 36, 165–192 (2019). https://doi.org/10.1007/s41412-019-00083-8

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