Proof.

Let σ−j indicate the strategies of all players in N∖j. For σ∈Hi;g, let σ′=σj′×σ−j and σjj′=1 producing σ′∈h−(i,σ;g). Let μjh=μh(j;σ)=|Nh(j;σ)| for h=x,y,s. For player j∈N∖i,

The payoff to j when leading is uncertain due to the uncertainty in the outcome of whether oi=oj.

The condition A(j;σ)≥0, derived from E(π(j;σ)−π(j;σ′))≥0, ensures that player j∈NS(i;σ) prefers her position as a follower of i over leading.

The first term of Aj;σ as expressed in eq. (8) is strictly positive. The coefficient on the second term is also positive. For θ=(m−1)rc/re>1 the third coefficient is also positive making it a sufficient condition for Aj;σ>0 for all j∈N∖i. The necessary and sufficient condition ensuring A(j;σ)≥0 for all j∈N∖i sets a lower threshold on θ, allowing the third term to be negative. For

or equivalently, θ≥1−1n−1, Aj;σ>0 for all j since μx(j)≤μx(jˉ)=n−2 and μy(j)≥μy(jˉ)=0.

Lemma 1.

Forσ,σ′∈Hi;g with σ−j=σ−j′and{aj,aj′}∈Nd(j;g), then forμx(j;σ)≤μx(j;σ′),

Lemma 2.

σ∈h′i;g is a necessary condition for σ∈hi;gto be a Nash equilibrium.

Proof.

For player i, leading dominates following since to choose one’s own successor as a predecessor pays zero. From μe(j)=μjy+μjs and μjy+μjs=μis−μjx, decreasing μjx increases πNL(j;σ) for any reward function that is increasing in μe. Among the following options, a player can do no better than to minimize μjx. A player who is not minimizing μjx is not optimizing against her available following options. Thus, any structure σ∈hi;g∖h′i;g cannot be a Nash equilibrium.

Proposition 2.

Given λ′(μ)≥0, {H′(i;g)}i∈N a set of equilibrium structures if and only if B≥0.

Proof.

From Proposition 1, given BNL≥0, every player j∈N∖i prefers any structure σ∈Hi;g over the structure produced by player j’s deviation to lead. In combination with Lemma 2, BNL≥0 implies that no follower in the population can do better for herself than to minimize her μjx.

Corollary 2.

Given λ′(μ)≥0, {H∗(i;g)}i∈N is a set of equilibrium structures if and only if BNL≥0.

Proof

H∗(i;g)⊆H′(i;g) implies that for σ∈Hi;g and aj∈Nd(j;g), if aj=argminNd(j;g)dji, then aj=argminNd(j;g)μjx. As further distinction between the strategies, structure σ′∈H′i;g if σ′∈H∗(i;g) or if σ′=σ−j×σj′ with aj′=j′ where σ∈H′i;g and where j∈NS(i;σ) satisfies the following three properties:

1. There exists j′∈Nd(j,g) with dj′i=dji, indicating that j′ is equidistant to the leader as is j and that j has the option to imitate j′,

2. μy(j;σ)=0, indicating that there are no successors to i of greater distance to i than j without also being a successor to j, and

3. either μs(j;σ)=0 or μs(j;σ)>0 with successors NS(j;σ) having no option to link to i but through j.

For {j1,j2}∈Nd(j;g) with dj1i<dj2i, let σh=σ|σjjh=1, h=1,2, so that μx(j,σ1)≤μx(j,σ2). The condition that allows μx(j,σ1)=μx(j,σ2) is μjy=0. With j2∉NS(j;σ), μjy=0 implies j2∈Nx(j;σ) and dj2i=dji=dj1i+1. For σ1∈H′(i;g), a necessary and sufficient condition to have σ2∈H′(i;g) is that for all js∈NS(j;σ1), Nd(js;g)⊂{NS(j;σ)∪{j}}. The condition establishes that no successor of j has the option to link to i without having the chain of links pass through j, a condition necessary to ensure that μx(js;σ2) is minimized for all js.

From H∗(i;g)⊆H′(i;g), σ∈H∗(i;g) is an equilibrium if BNL≥0 and σ∈H∗(i;g) is not an equilibrium if BNL<0.

Proposition 3.

For B<0 and σ∈hL(i,μis;g), all j∈NS(i;σ) prefer following i to leading and the remaining population, j∈N∖{i,NS(i;σ)}, prefer leading to following i if and only if σ∈hL∗(i,nˉ;g).

Proof.

Let μjh=μh(j;σ)=|Nh(j;σ)| for h=x,y,s and μl=μl(σ)=|NL(σ)|. For σ∈hL(i,μis;g), let σ′=σj′×σ−j and σjj′=1, j∈NS(i;σ). Uncertainty in the payoff to j when following stems from the uncertainty in whether oi=ol for each l∈NL(σ)∖{i} with,

The payoff to j when leading is uncertain due to the uncertainty in the outcome of whether oj=ol for each l∈NL(σ) with,

E(π(j;σ)−π(j;σ′)) from eqs. (24) and (25) is the same as from eqs. (22) and (23) when expressed in terms of μjx, μjy, and μjs as in eq. (8).^[25] The presence of a population of autonomous adopters does not alter the condition A(j;σ)≥0 for player j to prefer following to leading. Let jˉ(μis)=argmaxj∈NS(i;σ)dji, then μy(jˉ(μis))=μs(jˉ(μis))=0 and μx(jˉ(μis))=μis−1 so that

and C(μis;θ)=A(jˉ(μis);σ)m/reμis. With B<0, ((m−1)rc−re)<0 so that A(jˉ(μis);σ) decreases as the size of the tree increases. For μis=1, A(jˉ(1),σ)=(m−1)rc>0 while B<0 means that for μis=n−1, A(jˉ(n−1);σ)<0.

For m=1, A(j;σ)=−reμjx<0. For rc=0, A(j;σ)=re((m−1)μjy−μjx) so that the most distant follower, with A(jˉ(μis);σ)=−re(μis−1)≤0, prefers to lead in the presence of other followers. Player jˉ is indifferent to leading only when she is the only follower, m=1, and rc=0. With a non-trivial choice m<1 and a preference for conformity (rc>0), the equilibrium structure requires μis≥1.

The value of μis that sets C(μis;θ)=0 need not be an integer. There exists nˉ∈{floor(μ∗),ceil(μ∗)} such that A(j(nˉ);σ)≥0 and A(j(nˉ+1);σ)<0. A structure σ∈hL(i,nˉ;g)∖hL∗(i,nˉ;g) cannot be an equilibrium because either there are members of NS(i;σ) able to improve their payoff by choosing a different predecessor offering a shorter distance to i or there is a member of NL(σ) able to improve her payoff by choosing to follow a predecessor offering a shorter distance to i than the current jˉ(μis) player. For σ∈hL∗(i,nˉ;g), no player is able to improve her payoff through unilateral deviation while preserving a single-leader structure.

Proposition 4.

{hL∗(i,n‾;g)}i∈Nis the set of equilibrium strategies if and only if B < 0.

Proof.

Let μhs=μs(ih). For σ∈h∗(iA,iB;g), let σ′=σ−j×σj′, with σjj′=1, j∈NS(i;σ). For j, the expected payoff for following and leading are, respectively,

where μjh=μh(j;σ)=|Nh(j;σ)| for h=x,y,s,α,β and μl=μl(σ)=|NL(σ)|. Observe, for h=A,B,

The condition E(π(j;σ)−π(j;σ′))≥0 implies D(jh;σ)≥0 as reported in eq. (12). For the most distant player(s) from ih according to σ, E(ih;σ)=D(jˉ(μhs);σ)/reμhs. With μy(jˉ(μhs))=μs(jˉ(μhs))=0, μx(jˉ(μhs))=μhs−1, and μα(jˉ(μhs))≥μα(jh) for all jh∈NS(ih;σ), D(jˉ(μhs);σ)≥0 implies D(jh;σ)≥0 for all j∈NS(ih;σ), so that E(ih;σ)≥0 is necessary and sufficient to ensure D(jh;σ)≥0 holds for all jh∈NS(ih;σ). Since

the condition E(ih;σ)≥0 violates B>0. For σ∈hL∗(i,nˉ;g), no player is able to improve her payoff through unilateral deviation.

Proposition 5.

For

a structure σ∈H(iA,iB;g) is a Nash equilibrium if and only if σ∈H+(iA,iB;g). The set H+(iA,iB;g) is feasibly non-empty.