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Farsighted Clustering with Group-Size Effects and Reputations

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Abstract

We formulate a new model of strategic group formation by farsighted players in a seller–buyer setting. In each period, sellers are partitioned into groups/brands. At the end of each period, one seller may fail and exit the market by exogenous shock. When there is a vacant slot in the market, an entrant seller comes and chooses which existing group to join or to create a new group. There is a trade-off: larger groups enjoy more-than-proportional benefits of group size thanks to, for example, their visibility to attract customers and their negotiation power in factor markets. However, larger groups are more likely to experience member failure, which is a reputation loss. We find that when the rate of reputation loss is small, clustering is inevitable, but as the rate of reputation loss increases, the largest group with a bad reputation does not attract an entrant, dissolving a cluster. With a limited group-size benefit and a high rate of reputation loss, all entrants create a new group; that is, no clustering occurs. A mathematically interesting result is that, even though the model itself is stationary and symmetric, depending on the parameters, there may be multiple pure-strategy, symmetric stationary equilibria, or there may be no such equilibrium. The economic implications include that group reputation may prevent clustering and that similar markets can have different cluster structures.

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Notes

  1. To focus on the novel trade-off, we do not include the option of existing group members refusing or requesting an entrant to join, an entrant’s cost of joining an existing group or creating a new group, and so on. Empirical evidence [7] suggests that a group can be just similar names, and in such cases, we do not need to consider agreement/cost to join a group. We also do not consider information transmission from surviving consumers to newcomers (cf. [4]). Our reputation externality from a failed seller to other members in the same group is also different from alliance formation externalities. See the surveys by Bloch [3] and Yi [17].

  2. The four-seller case is similar but tedious to write out. A general analysis is quite difficult, which will be explained in the text.

  3. In the Online Appendix, we present a cyclic equilibrium for a parameter combination in which no pure-strategy stationary equilibrium exists.

  4. If \(N=2\), then we cannot address the more-than-proportional, group-size effect.

  5. Although we do not address each consumer’s optimization in this paper, if consumers are also active players, \(\delta \) is their effective discount factor (e.g., [7]).

  6. In Japan, there have been many incidents where workers posted a scandalous video taken at the workplace on the social networks, for example, spitting in the kitchen pots or lying down in an ice cream freezer. Some shops were forced to go out of business because of such posts.

  7. Alternatively, we can adapt the model of [7] to the current environment so that consumers also choose sellers strategically. If all sellers’ products (and prices) are the same on the equilibrium path, then the behavior in Assumption 1 is an optimal one for consumers.

  8. If the relevant seller belongs to the monopoly group, i.e., \(g_1=N\), then \(i\ne 1\) does not exist.

  9. From the viewpoint of an entrant, interim states \(1_F1\) and \(11_F\) are the same.

  10. However, there may be nonstationary Markov equilibria in which entrants take into account both the interim state and the period in the game. See the Online Appendix.

  11. Since we omit the state “exit”, T is not a probability matrix. Each row sums to \(1-\frac{\varepsilon }{3}\).

  12. This formula essentially corresponds to Equation (1) of [11]. However, we omit the state “exit” so that T is not the probability matrix, as noted in the above Footnote 11, and the effective discount factor based on the exit rate \(\varepsilon /3\) is embedded in T.

  13. Similar figures for all \(k>1\) are available upon request.

  14. We also have similar results for \(N=4\).

  15. A variable seller-population model where N changes over time can also be formulated if there is a fixed upper bound to N.

  16. Since a Nash equilibrium holds under weak inequalities, we did not partition the cases. Thus the “either” statement is not an exclusive one.

  17. We solved the system of equations in two ways: by using Mathematica for the recursive equation (6) and by hand. The detailed calculations (and the Mathematica codes) are available from the authors on request.

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Acknowledgements

We thank two anonymous reviewers for very useful comments that improved the paper. We also thank Daisy Weijia Dai, who gave us numerous useful references.

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Correspondence to Takako Fujiwara-Greve.

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This article is part of the topical collection ‘Group Formation and Farsightedness” edited by Francis Bloch, Ana Mauleon and Vincent Vannetelbosch.

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Appendix

Appendix

1.1 A.1. Optimal Value Functions and the Proof of Lemma 1

The stationary relationships of the optimal value functions at all states \(s \in S_3\) other than (2), (3), (4) are as follows.

$$\begin{aligned} V_t(12)= & {} U(12) + \frac{2\varepsilon }{3}V_{t+1}(21_F) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F) \geqq V_{t+1} (2_F1), V_{t+1}(21_F) \geqq V_{t+1}(11_F1) \bigr )\\&+ \frac{2\varepsilon }{3}V_{t+1}(12_F) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1} (2_F1), V_{t+1}(2_F1) \geqq V_{t+1}(11_F1) \bigr )\\&+ \frac{2\varepsilon }{3}V_{t+1}(11_F1) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1} (11_F1), V_{t+1}(2_F1)< V_{t+1}(11_F1) \bigr )\\&+ (1-\varepsilon ) V_{t+1}(12),\\ V_t(12_F)= & {} U(12_F) + \frac{2\varepsilon }{3}V_{t+1}(21_F) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F) \geqq V_{t+1} (2_F1), V_{t+1}(21_F)\geqq V_{t+1}(11_F1) \bigr )\\&+ \frac{2\varepsilon }{3}V_{t+1}(12_F) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1} (2_F1), V_{t+1}(2_F1)\geqq V_{t+1}(11_F1) \bigr )\\&+ \frac{2\varepsilon }{3}V_{t+1}(11_F1) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1} (11_F1), V_{t+1}(2_F1)< V_{t+1}(11_F1) \bigr )\\&+ (1-\varepsilon ) V_{t+1}(12),\\ V_t(1_F2)= & {} U(1_F2) + \frac{2\varepsilon }{3}V_{t+1}(21_F) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F) \geqq V_{t+1} (2_F1), V(21_F)\geqq V_{t+1}(11_F1) \bigr )\\&+ \frac{2\varepsilon }{3}V_{t+1}(12_F) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1} (2_F1), V_{t+1}(2_F1)\geqq V_{t+1}(11_F1) \bigr )\\&+ \frac{2\varepsilon }{3}V_{t+1}(11_F1) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1} (11_F1), V_{t+1}(2_F1)< V_{t+1}(11_F1) \bigr )\\&+ (1-\varepsilon ) V_{t+1}(12),\\ V_t(21)= & {} U(21) +\frac{\varepsilon }{3} V(3)\cdot \texttt {IF} \bigl ( V_{t+1}(3) \geqq V_{t+1} (12)\bigr ) +\frac{\varepsilon }{3} V_{t+1}(21)\cdot \texttt {IF} \bigl ( V_{t+1}(3)< V_{t+1} (12)\bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(1_F2) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F) \geqq V_{t+1}(2_F1), V_{t+1}(21_F) \geqq V_{t+1}(11_F1) \bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(2_F1) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1}(2_F1), V_{t+1}(2_F1) \geqq V_{t+1}(11_F1) \bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(1_F11) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1}(11_F1), V_{t+1}(2_F1)< V_{t+1}(11_F1) \bigr )\\&+ (1-\varepsilon ) V_{t+1}(21),\\ V_t(2_F1)= & {} U(2_F1) +\frac{\varepsilon }{3} V_{t+1}(3)\cdot \texttt {IF} \bigl ( V_{t+1}(3) \geqq V_{t+1} (12)\bigr ) \\&+\frac{\varepsilon }{3} V_{t+1}(21)\cdot \texttt {IF} \bigl ( V_{t+1}(3)< V_{t+1} (12)\bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(1_F2) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F) \geqq V_{t+1}(2_F1), V_{t+1}(21_F) \geqq V_{t+1}(11_F1) \bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(2_F1) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1}(2_F1), V_{t+1}(2_F1) \geqq V_{t+1}(11_F1) \bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(1_F11) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1}(11_F1), V_{t+1}(2_F1)< V_{t+1}(11_F1) \bigr )\\&+ (1-\varepsilon ) V_{t+1}(21),\\ V_t(21_F)= & {} U(21_F) +\frac{\varepsilon }{3} V_{t+1}(3)\cdot \texttt {IF} \bigl ( V_{t+1}(3) \geqq V_{t+1} (12)\bigr ) \\&+\frac{\varepsilon }{3} V_{t+1}(21)\cdot \texttt {IF} \bigl ( V_{t+1}(3)< V_{t+1} (12)\bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(1_F2) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F) \geqq V_{t+1}(2_F1), V_{t+1}(21_F) \geqq V_{t+1}(11_F1) \bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(2_F1) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1}(2_F1), V_{t+1}(2_F1) \geqq V_{t+1}(11_F1) \bigr )\\&+\frac{\varepsilon }{3} V_{t+1}(1_F11) \cdot \texttt {AND}\bigl ( V_{t+1}(21_F)< V_{t+1}(11_F1), V_{t+1}(2_F1)< V_{t+1}(11_F1) \bigr )\\&+ (1-\varepsilon ) V_{t+1}(21),\\ V_t(3)= & {} U(3) +\frac{2\varepsilon }{3} V_{t+1}(2_F1) \cdot \texttt {IF} \bigl ( V_{t+1} (12_F) \geqq V_{t+1}(3_F) \bigr )\\&+\frac{2\varepsilon }{3} V_{t+1}(3_F) \cdot \texttt {IF} \bigl ( V_{t+1} (12_F)< V_{t+1}(3_F) \bigr ) +(1-\varepsilon ) V_{t+1}(3),\\ V_t(3_F)= & {} U(3_F) +\frac{2\varepsilon }{3} V_{t+1}(2_F1) \cdot \texttt {IF} \bigl ( V_{t+1} (12_F) \geqq V_{t+1}(3_F) \bigr )\\&+\frac{2\varepsilon }{3} V_{t+1}(3_F) \cdot \texttt {IF} \bigl ( V_{t+1} (12_F) < V_{t+1}(3_F) \bigr ) +(1-\varepsilon ) V_{t+1}(3), \end{aligned}$$

where AND is a function such that for any two statements A and B, AND\((A,B)=1\) if both A and B are true, and AND\((A,B)=0\) otherwise.

Proof of Lemma 1

We can simplify some of the above equations.

$$\begin{aligned} V_t(1_F11)&= V_t(111)-U(111) + U(1_F11),\\ V_t(11_F1)&= V_t(111)-U(111) + U(11_F1),\\ V_t(12_F)&= V_t(12)-U(12) + U(12_F),\\ V_t(1_F2)&= V_t(12)-U(12) + U(1_F2),\\ V_t(2_F1)&= V_t(21)-U(21)+U(2_F1), \\ V_t(21_F)&= V_t(21)-U(21)+ U(21_F), \\ V_t(3_F)&= V_t(3)-U(3)+ U(3_F) = V_t(3). \end{aligned}$$

Hence (dropping the time subscript),

$$\begin{aligned} V(21_F)-V(2_F1)= & {} U(21_F)-U(2_F1) \ = \ \frac{\delta }{2} \ > \ 0. \end{aligned}$$

We also have

$$\begin{aligned} V(21_F)-V(11_F1)= & {} V(21)-U(21)+U(21_F)-V(111)+U(111)-U(11_F1)\\= & {} V(21)-V(111) -\frac{2^{k-1}\delta }{1+2^k}+\frac{\delta }{2}-\frac{\delta }{6}\\= & {} V(21)-V(111) + \frac{(1-2^{k-1})\delta }{3(1+2^k)}. \end{aligned}$$

Since \(k>1\), \(V(21_F) \geqq V(11_F1)\) implies \(V(21) > V(111)\). Similarly,

$$\begin{aligned} V(12_F) -V(3_F)= & {} V(12)-U(12)+U(12_F) - V(3)\\= & {} V(12)-V(3) +\frac{2^{k-1}\delta }{1+2^k}. \end{aligned}$$

Thus, \(V(12) \geqq V(3)\) implies \(V(12_F) > V(3_F)\). \(\square \)

In view of Lemma 1, we have the classification of possible relationships of value functions as in Table 4.Footnote 16 Based on this classification, we have only nine possible strategies as candidates of symmetric, stationary equilibrium strategies as listed in Table 2.

Table 4 Possible relationships of long-run values

1.2 A.2 Derivation of the f1-Equilibrium

We only show the detailed computations for the equilibrium of the “Join-the-largest-without-F” strategy (f1). The derivation of the bounds for other strategies as well as the relationships among the bounds are given in Online Appendix.

Suppose that

$$\begin{aligned} V(21) \geqq V(111),\ V(21_F) \geqq V(11_F1),\ V(3) \geqq V(12), \ \mathrm{and} \ V(12_F) \geqq V(3_F). \end{aligned}$$

Then the equations that the value function satisfies (shown in the text and in Appendix, Sect. A.1) become

$$\begin{aligned} V(111)= & {} U(111) + \frac{2\varepsilon }{3} \left\{ \frac{1}{2}V(21)+\frac{1}{2}V(12)\right\} +(1-\varepsilon ) V(111),\\ V(1_F11)= & {} V(111)-U(111) + U(1_F11),\\ V(11_F1)= & {} V(111)-U(111) + U(11_F1),\\ V_(12)= & {} U(12) + \frac{2\varepsilon }{3}V(21_F) + (1-\varepsilon ) V(12),\\ V(12_F)= & {} V(12)-U(12) + U(12_F),\\ V(1_F2)= & {} V(12)-U(12) + U(1_F2), \\ V(21)= & {} U(21) +\frac{\varepsilon }{3} V(3) +\frac{\varepsilon }{3} V(1_F2) + (1-\varepsilon ) V(21),\\ V(2_F1)= & {} V(21)-U(21)+U(2_F1), \\ V(21_F)= & {} V(21)-U(21)+ U(21_F), \\ V(3)= & {} U(3) +\frac{2\varepsilon }{3} V(2_F1) +(1-\varepsilon ) V(3),\\ V(3_F)= & {} V(3). \end{aligned}$$

We have the following solution for this system of equations:Footnote 17

$$\begin{aligned} V(21)= & {} \frac{4-2\delta \varepsilon + 11 \cdot 2^{k-1}-2^k \delta \varepsilon }{5(1+2^k) \varepsilon },\\ V(2_F1)= & {} \frac{4-2\delta \varepsilon + 11 \cdot 2^{k-1}-7 \cdot 2^{k-1} \delta \varepsilon }{5(1+2^k)\varepsilon },\\ V(21_F)= & {} \frac{8+\delta \varepsilon + 11 \cdot 2^{k}-2^{k+1} \delta \varepsilon }{10(1+2^k)\varepsilon },\\ V(12)= & {} \frac{23+\delta \varepsilon + 11 \cdot 2^{k}-2^{k+1}\delta \varepsilon }{15(1+2^k)\varepsilon },\\ V(12_F)= & {} \frac{23+\delta \varepsilon + 11 \cdot 2^{k}+13\cdot 2^{k}\delta \varepsilon }{15(1+2^k)\varepsilon }, \\ V(1_F2)= & {} \frac{23-14\delta \varepsilon + 11 \cdot 2^{k}-2^{k+1}\delta \varepsilon }{15(1+2^k)\varepsilon }, \\ V(111)= & {} \frac{10-\delta \varepsilon + 17 \cdot 2^{k-1}-2^k \delta \varepsilon }{9(1+2^k)\varepsilon },\\ V(1_F11)= & {} \frac{10-4\delta \varepsilon + 17 \cdot 2^{k-1}-2^{k+2} \delta \varepsilon }{9(1+2^k)\varepsilon },\\ V(11_F1)= & {} \frac{20+\delta \varepsilon + 17 \cdot 2^{k}+2^{k}\delta \varepsilon }{18(1+2^k)\varepsilon },\\ V(3)= & {} \frac{13-4\delta \varepsilon + 2^{k+4}-7\cdot 2^{k}\delta \varepsilon }{15(1+2^k)\varepsilon },\\ V(3_F)= & {} \frac{13-4\delta \varepsilon + 2^{k+4}-7\cdot 2^{k}\delta \varepsilon }{15(1+2^k)\varepsilon }. \end{aligned}$$

Note that

$$\begin{aligned} V(21_F)-V(11_F1) =&\ \frac{8+\delta \varepsilon +11 \cdot 2^{k}-2^{k+1}\delta \varepsilon }{10(1+2^k)\varepsilon } -\frac{20+\delta \varepsilon +17\cdot 2^{k}+2^k\delta \varepsilon }{18(1+2^k)\varepsilon }\\ =&\ \frac{-28+4\delta \varepsilon + 14\cdot 2^{k}-23 \cdot 2^{k}\delta \varepsilon }{90(1+2^k)\varepsilon },\\ V(3)-V(12) =&\ \frac{13-4\delta \varepsilon + 2^{k+4}-7\cdot 2^k \delta \varepsilon }{15(1+2^k)\varepsilon } -\frac{23+\delta \varepsilon + 11\cdot 2^k-2^{k+1}\delta \varepsilon }{15(1+2^k)\varepsilon } \\ =&\ \frac{-2-\delta \varepsilon +2^k -2^k\delta \varepsilon }{3(1+2^k)\varepsilon }. \end{aligned}$$

Thus,

$$\begin{aligned} V(21_F) \geqq V(11_F1)\Leftrightarrow & {} \delta \varepsilon \leqq \frac{28(2^{k-1}-1)}{23 \cdot 2^{k}-4},\\ V(3) \geqq V(12)\Leftrightarrow & {} \delta \varepsilon \leqq \frac{2 (2^{k-1}-1)}{2^k+1}. \end{aligned}$$

Since

$$\begin{aligned} \frac{2}{2^k+1}-\frac{28}{23\cdot 2^{k}-4}= & {} \frac{36(2^{k-1}-1)}{(2^k+1)(23\cdot 2^{k}-4} \ > \ 0, \end{aligned}$$

we have

$$\begin{aligned} V(21_F) \geqq V(11_F1) \text { and } V(2) \geqq V(12)\Leftrightarrow & {} \delta \varepsilon \leqq \frac{28(2^{k-1}-1)}{23\cdot 2^k-4} . \end{aligned}$$

Note also that

$$\begin{aligned} V(12_F)-V(3_F)= & {} \frac{23+\delta \varepsilon + 11\cdot 2^k + 13 \cdot 2^k \delta \varepsilon }{15(1+2^k)\varepsilon } -\frac{13-4\delta \varepsilon +2^{k+4}-7\cdot 2^k\delta \varepsilon }{15(1+2^k)\varepsilon } \\= & {} \frac{2+\delta -2^k+2^{k+2}\delta \varepsilon }{3(1+2^k)\varepsilon }. \end{aligned}$$

Thus,

$$\begin{aligned} V(12_F) \geqq V(3_F)\Leftrightarrow & {} \delta \varepsilon \geqq \frac{2(2^{k-1}-1)}{2^{k+2}+1}. \end{aligned}$$

Since

$$\begin{aligned} \frac{28}{23\cdot 2^k-4}-\frac{2}{2^{k+2}+1}= & {} \frac{33\cdot 2^{k+1}+36}{(23\cdot 2^k-4)(2^{k+2}+1)} \ > \ 0, \end{aligned}$$

we have

$$\begin{aligned} \left. \begin{array}{rcl} V(21) &{}\geqq &{} V(111)\\ V(21_F) &{}\geqq &{} V(11_F1) \\ V(3) &{}\geqq &{} V(12)\\ V(12_F) &{}\geqq &{} V(3_F) \end{array} \right\}\Leftrightarrow & {} {\underline{x}}_1 \leqq \delta \varepsilon \leqq {\overline{x}}_1, \end{aligned}$$

where

$$\begin{aligned} {\underline{x}}_1:= & {} \frac{2(2^{k-1}-1)}{2^{k+2}+1},\\ {\overline{x}}_1:= & {} \frac{28(2^{k-1}-1)}{23 \cdot 2^k-4}. \end{aligned}$$

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Fujiwara-Greve, T., Hokari, T. Farsighted Clustering with Group-Size Effects and Reputations. Dyn Games Appl 13, 610–635 (2023). https://doi.org/10.1007/s13235-022-00472-w

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