Symmetric Equilibria in a Cost-Averting War of Attrition Requiring Minimum Necessary Conceders

Geofferey Jiyun Kim; Bara Kim

doi:10.1515/bejte-2016-0109

Article

Symmetric Equilibria in a Cost-Averting War of Attrition Requiring Minimum Necessary Conceders

Geofferey Jiyun Kim and Bara Kim

Published/Copyright: September 26, 2017

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From the journal The B.E. Journal of Theoretical Economics Volume 18 Issue 1

Abstract

This paper provides an analysis of a cost-averting war of attrition with minimum necessary conceders. All symmetric stationary Nash equilibria are characterized. The multiplicity of equilibria has called for further refinements. We show that there exists a unique symmetric stationary trembling hand perfect equilibrium. Comparative statics results of the trembling hand perfect equilibrium are provided. This paper’s model is motivated by the problem of delayed public goods provisions in collective action settings. Augmenting the number of minimum necessary conceders can curtail delays.

Keywords: Bellman equation; Nash equilibrium; public economics; trembling hand perfection; war of attrition

JEL Classification: C7

Appendix

A Proof of Lemma 3

Equation (6) is written as

(12)g(p)=C∑k=n−1N−1b(k;N−1,p)b(n−1;N−1,p)1k+1−κ1−dB(n−1;N−1,p),

where

g(p)≡h(p)1−dB(n−2;N−1,p)b(n−1;N−1,p).

Since b(k;N−1,p)b(n−1;N−1,p) is strictly increasing in p on (0,1) for k≥n and B(n−1;N−1,p) is strictly decreasing in p on (0,1), g(p) is strictly increasing in p on (0,1).

Hence, exactly one of the following holds: (i) limp→0+g(p)<0<limp→1−g(p), (ii) limp→0+g(p)≥0, and (iii) limp→1−g(p)≤0. Since h(1)=CN>0 and 1−dB(n−2;N−1,p)b(n−1;N−1,p)>0 for all p∈(0,1), the last case is impossible and the remaining two cases are written as following:

there exists p~∈(0,1) such that h(p)<0 for p∈(0,p~) and h(p)>0 for p∈(p~,1);
h(p)>0 for all p∈(0,1).

By direct calculation,

(13)h(k)(0)={0if k=0,…,n−2,11−d(N−1)!(N−n)!(Cn−κ1−d)if k=n−1,(N−1)(N−2)3(1−d)−(N−2+d(N−1))h(1)(0)if k=n=2,11−d(N−1)!(N−n−1)!1n+1−(N−n)h(n−1)(0)if k=n≥3,

where h(k) denotes the kth order derivative of the function h. By eq. (13), if Cn<κ1−d, then h(0)=0 and h(p)<0 for sufficiently small positive p. Hence, if Cn<κ1−d, then (a) holds. We have proved the first assertion of Lemma 3. By (13), if Cn≥κ1−d, then h(0)=0 and h(p)>0 for sufficiently small positive p. Hence, if Cn≥κ1−d, then (b) holds. We have proved the second assertion of Lemma 3.

B Proofs of Corollaries 1, 2 and 3, and (11)

The first assertion of each corollary follows from the second assertion of Theorem 1. We will prove the second assertion of each corollary. Let us use g(p,C,κ,n) for the right hand side of eq. (12), instead of g(p), to explicitly express the dependence of C, κ and c.

Proof of Corollary 1

Suppose n>(1−d)Cκ. Since

g(p,C,κ,n)−g(p,C,κ,n+1)=C∑k=n−1N−2(b(k;N−1,p)b(n−1;N−1,p)1k+1−b(k+1;N−1,p)b(n;N−1,p)1k+2)+Cb(N−1;N−1,p)b(n−1;N−1,p)1N−κ(11−dB(n−1;N−1,p)−11−dB(n;N−1,p))=C∑k=n−1N−2(p1−p)k−n+1((n−1)!(N−n)!(k+1)!(N−k−1)!−n!(N−n−1)!(k+2)!(N−k−2)!)+Cb(N−1;N−1,p)b(n−1;N−1,p)1N+κdb(n;N−1,p)(1−dB(n−1;N−1,p))(1−dB(n;N−1,p))=C∑k=n−1N−2(p1−p)k−n+1(n−1)!(N−n−1)!(N(k−n+2)−n)(k+2)!(N−k−1)!+Cb(N−1;N−1,p)b(n−1;N−1,p)1N+κdb(n;N−1,p)(1−dB(n−1;N−1,p))(1−dB(n;N−1,p)),

g(p,C,κ,n)−g(p,C,κ,n+1)is positive. Therefore g(p,C,κ,n) is strictly decreasing in n. We know that g(p,C,κ,n) is striclty increasing in p. Hence p∗, which is the unique zero of g(p,C,κ,n) on (0,1), is strictly increasing in n. By eq. (8), ρ∗ is strictly increasing in n. By eq. (9), E[τ∗] is strictly decreasing in n. Equation (10) is written as

c∗=CN−κd+1−dd(κ1−d−CN)1−d(1−ρ∗).

Since ρ∗ is strictly increasing in n, c∗ is strictly decreasing in n.

Proof of Corollary 2

Suppose C<nκ1−d. Since p∗ is the unique zero of g(p,C,κ,n) on (0,1),

∂p∗∂C=−∂∂Cg(p∗,C,κ,n)∂∂pg(p∗,C,κ,n).

By direct calculation, it is easy to see that ∂∂Cg(p∗,C,κ,n)>0 and ∂∂pg(p∗,C,κ,n)>0. Hence, ∂p∗∂C<0, i.e., p∗ is strictly decreasing in C. By eq. (8), ρ∗ is strictly decreasing in C. By eq. (9), E[τ∗] is strictly increasing in C. By eq. (10),

∂c∗∂C=ρ∗/N1−d(1−ρ∗)−(1−d)(κ1−d−CN)(1−d(1−ρ∗))2∂ρ∗∂C,

which is positive. Hence, c∗ is strictly increasing in C.

Proof of Corollary 3

Suppose κ>(1−d)Cn. Since ∂∂κg(p∗,C,κ,n)<0 and ∂∂pg(p∗,C,κ,n)>0,

∂p∗∂κ=−∂∂κg(p∗,C,κ,n)∂∂pg(p∗,C,κ,n)>0,

i.e., p∗ is strictly increasing in κ. By eq. (8), ρ∗ is strictly increasing in κ. By eq. (9), E[τ∗] is strictly decreasing in κ.

Proof of (11)

Since p∗ is the unique zero of g(p,C,κ,n) on (0,1),

(14)C∑k=n−1N−1b(k;N−1,p∗)b(n−1;N−1,p∗)1k+1=κ1−dB(n−1;N−1,p∗).

By eqs. (8) and (14),

limκ→∞κ(1−ρ∗)=Climκ→∞∑k=n−1N−1b(k;N−1,p∗)B(n−1;N,p∗)b(n−1;N−1,p∗)1−dB(n−1;N−1,p∗)k+1=0.

Acknowledgements:

We would like to thank the anonymous reviewer for helpful comments and suggestions, which immensely improved this article.

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Published Online: 2017-9-26

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https://doi.org/10.1515/bejte-2016-0109

Keywords for this article

Bellman equation; Nash equilibrium; public economics; trembling hand perfection; war of attrition