Neighborhood structure and the evolution of cooperation

Abstract

This paper deals with the problem of explaining the survival of cooperative behavior in populations in which each person interacts only with a small set of social ‘neighbors’, and individuals adjust their behavior over time by myopically imitating more successful strategies within their own neighborhood. We identify two parameters—the interaction radius and the benefit–cost ratio—which jointly determine whether or not cooperation can survive. For each value of the interaction radius, there exists a critical value of the benefit–cost ratio which serves as the threshold below which cooperation cannot be sustained. This threshold itself declines as the interaction radius rises, so there is a precise sense in which dense networks are more conducive to the evolution of cooperation.

Appendix

Proof of Proposition 1

We shall prove that i s(0) ∈ L(3r,3r) and β > β _h (r) then there exists τ such that for all t ≥ τ, s(t) ∈ L(3r,0) ∩ U(n,3r), from which the result follows. We begin with the following results (the first two of which follow from the discussion in the text)||□

Lemma 1

Suppose s(t) ∈ L(3r, 2r). Then $s_{i}(t)=1\Rightarrow s_{i}(t+1)=1$ (all altruists remain altruists) if and only if β > β _h (r).

Lemma 2

Suppose s(t) ∈  L(2r,3r). Then s _i (t) = 0 and $\sum_{j=i-r}^{i+r}s_{j}(t)\neq 0\Rightarrow s_{i}(t+1)=1$ (all egoists with one or more altruist neighbors become altruists) if and only if β > β _h (r).

Lemma 3

Suppose that β > β _h (r) and s(t)  ∈ L(3r,2r + 1). Then all altruist clusters expand and all egoist clusters contract.

Proof of Lemma 3

From Lemma 1, all altruist clusters expand or survive. From Lemma 2, all egoist clusters of length at least 3r contract. If we can show that all egoist clusters of length l with 2r + 1 ≤ l ≤ 3r − 1 also contract then it follows from Lemma 1 that all altruist clusters expand. Accordingly, let the players {1,...,l} constitute such a cluster. By hypothesis, this cluster must be adjacent (on either side) to an altruist cluster of length at least 3r. Exactly l − 2r > 0 players in the egoist cluster {1,...,l} are in homogeneous egoist neighborhoods and thus will not switch. To see which, if any, of the remaining egoists switch, consider the egoist players m, where m ∈ {1,...,r}. From the perspective of player m, the average altruist payoff $\bar{\pi}_{m}^{a}$ is exactly as given in Eq. 10 in the text:

$$ \bar{\pi}_{m}^{a}=-1+\frac{1}{4}\left( 3r-m\right) \frac{\beta }{r} $$

The average egoist payoff $\bar{\pi}_{m}^{e}$ observed by player m depends on whether or not m > l − 2r. For all players m ∈ {1,...,l − 2r}, $\bar{\pi} _{m}^{e}$ is exactly as given in Eq. 9 in the text, and hence (using the same argument as in the text), since β > β _h (r) all these egoists will switch to altruists. For m ∈ {l − 2r + 1,...,r}, on the other hand, the average egoist payoff from the perspective of player m is

$$\begin{gathered} \overline \pi _m^e = \left( {\frac{1}{{r + m}}} \right)\frac{\beta }{{2r}}\left( {\sum\limits_{i = 1}^r {\left( {r + 1 - i} \right) + } \sum\limits_{i = 1}^{l - 2r} {0 + } \sum\limits_{i = 1}^{m + 2r - l} i } \right) \\ = \frac{1}{4}\beta \frac{{5r^2 + 3r + m^2 + 4mr - 2ml + m - 4rl + l^2 - l}}{{\left( {r + m} \right)r}} \\ \end{gathered} $$

(12)

The average payoff difference is

$$\begin{array}{*{20}c} {\overline \pi _m^a - \overline \pi _m^e } \hfill \\ {\,\,\,\,\,\,\, = - 1 + \frac{1}{4}\left( {3r - m} \right)\frac{\beta }{r} - \frac{1}{4}\beta \frac{{5r^2 + 3r + m^2 + 4mr - 2ml + m - 4rl + l^2 - l}}{{\left( {r + m} \right)r}}} \hfill \\ {\,\,\,\,\,\, = - \frac{1}{4}\beta \frac{{2r^2 + 2mr + 2m^2 + 3r - 2ml + m - 4rl + l^2 - l}}{{\left( {r + m} \right)r}}} \hfill \\ \end{array} $$

Solution: Differentiating with respect to m and simplifying yields

$$ \frac{\partial \left( \bar{\pi}_{m}^{a}-\bar{\pi}_{m}^{e}\right) }{\partial m }=-\frac{1}{4}\beta \frac{4mr+2m^{2}+2rl-2r-l^{2}+l}{\left( r+m\right) ^{2}r} $$

Since we are considering the case m ∈ {l − 2r + 1,...,r}, we have l ≤ m + 2r − 1. Hence, provided that 2r ≤ l, the numerator of the above expression is

$$\begin{array}{*{20}c} {4mr + 2m^2 + 2rl - 2r - l^2 + l} \hfill \\ {\,\,\,\,\, \geqslant 4mr + 2m^2 + 2r\left( {2r} \right) - 2r - \left( {m + 2r - 1} \right)^2 + 2r} \hfill \\ {\,\,\,\,\, = m^2 + 2m + 4r - 1 > 0.} \hfill \\ \end{array} $$

Therefore $\bar{\pi}_{m}^{a}-\bar{\pi}_{m}^{e}$ is strictly decreasing in m and if egoist m switches, then so does egoist m − 1 for all m ∈ {l − 2r + 2,...,r}. Together with the fact that players {1,...,l − 2r} switch to altruists, this proves that the egoist cluster contracts. Hence from Lemma 1 both adjacent altruist clusters expand. Note that since l > 2r, the egoist cluster shrinks to a length strictly less than 2r in state s(t + 1).□

Lemma 4

Suppose that β > β _h (r) and s ^t  ∈ L(3r,2r) . Then all altruist clusters survive or expand and all egoist clusters survive, contract, or vanish.

Proof of Lemma 4

From Lemma 3, all egoist clusters of length greater than 2r contract so we need only consider clusters of length 2r. Let the players {1,...,2r} constitute such a cluster. By hypothesis, this cluster must be adjacent (on either side) to an altruist cluster of length at least 3r. Consider the egoist player m, where m ∈ {1,...,r} . From the perspective of player m, the average altruist payoff $\bar{\pi}_{m}^{a}$ is exactly as given in Eq. 10 above and the average egoist payoff is exactly as given in Eq. 12 above. As in the proof of Lemma 3 therefore, if any egoist switches then so do all egoists which are closer to an altruist cluster. Hence the egoist cluster cannot be punctured. Since all altruist clusters survive or expand from Lemma 1, the egoist cluster cannot expand. It must therefore survive, contract or vanish.□

Lemma 5

Suppose that s ^t  ∈ L(3r,r + 1). Then no new cluster appears, and all egoist cluster of length less than 3r continue to remain of length less than 3r.

Proof of Lemma 5

To show that no new cluster appears, we need to show that no cluster is punctured, and that two heterogeneous adjacent players cannot both switch. Given the above results, we need only consider egoist clusters of length l where r + 1 ≤ l ≤ 2r − 1, and their adjacent altruist clusters. Without loss of generality, let the players {1,...,l} constitute an egoist cluster with r + 1 ≤ l ≤ 2r − 1. By hypothesis, this cluster must be adjacent (on either side) to an altruist cluster of length at least 3r.□

Claim 1

For any m ∈ {l − r,...,r}, $s_{m+1}^{t+1}=0$ if $s_{m}^{t+1}=0$.

Proof of Claim 1

All (egoist)players m with m ∈ {l + 1 − r,...,r} have the same payoffs since all have exactly l egoist neighbors. They also observe the same mean egoist payoff since they observe all egoists in the cluster. The mean altruist payoff they observe is greater for values of m closer to the boundary of the cluster. Hence if player r does not switch, neither do any of the players {l + 1 − r,...,r}. Player l − r observes a higher mean altruist payoff than player l + 1 − r, and the same mean egoist payoff. Hence if l − r switches, l + 1 − r must switch.□

Claim 2

For any m ∈ {1,...,l − r − 1}, $s_{m+1}^{t+1}=0$ if $s_{m}^{t+1}=0$.

Proof of Claim 2

Consider player m with m ∈ {1,...,l − r}.

$$\begin{array}{*{20}c} {\overline \pi _m^e \left( {s^t } \right)} \hfill \\ {\,\,\,\,\, = \,\,\frac{1}{{r + m}}\left( {\sum\limits_{i = 1}^{l - r} {\frac{{\beta \left( {r + 1 - i} \right)}}{{2r}} + \frac{{\left( {2r - l} \right)\beta \left( {2r + 1 - l} \right)}}{{2r}} + } \sum\limits_{i = 1}^m {\frac{{\beta \left( {2r - l + i} \right)}}{{2r}}} } \right)} \hfill \\ {\overline \pi _{m + 1}^e \left( {s^t } \right)} \hfill \\ {\,\,\,\,\, = \,\,\frac{1}{{r + m + 1}}\left( {\sum\limits_{i = 1}^{l - r} {\frac{{\beta \left( {r + 1 - i} \right)}}{{2r}} + \frac{{\left( {2r - l} \right)\beta \left( {2r + 1 - l} \right)}}{{2r}} + } \sum\limits_{i = 1}^{m + 1} {\frac{{\beta \left( {2r - l + i} \right)}}{{2r}}} } \right)} \hfill \\ \end{array} $$

and

$$\begin{array}{*{20}c} {\overline \pi _m^a \left( {s^t } \right) = \frac{1}{{r + 1 - m}}\sum\limits_{i = 1}^{r + 1 - m} {\frac{{\beta \left( {r + i - 1} \right)}}{{2r}} - 1} } \\ {\overline \pi _{m + 1}^a \left( {s^t } \right) = \frac{1}{{r - m}}\sum\limits_{i = 1}^{r - m} {\frac{{\beta \left( {r + i - 1} \right)}}{{2r}} - 1} } \\ \end{array} $$

Now suppose that $\bar{\pi}_{m}^{e}(s^{t})-\bar{\pi}_{m}^{a}(s^{t})\geq 0$ (Player m remains an egoist). Then

$$\begin{array}{*{20}c} {\overline \pi _{m + 1}^e \left( {s^t } \right) - \overline \pi _{m + 1}^a \left( {s^t } \right)} \hfill \\ {\,\,\,\,\,\, \geqslant \left( {\overline \pi _{m + 1}^e \left( {s^t } \right) - \overline \pi _m^e \left( {s^t } \right)} \right) - \left( {\overline \pi _{m + 1}^a \left( {s^t } \right) - \overline \pi _m^a \left( {s^t } \right)} \right)} \hfill \\ {\,\,\,\,\, = \left( { - \frac{1}{4}\beta \frac{{r - l - m + l^2 - 2rm - m^2 - 2rl + r^2 }}{{\left( {r + m + 1} \right)r\left( {r + m} \right)}}} \right) - \left( { - \frac{1}{4}\frac{\beta }{r}} \right)} \hfill \\ {\,\,\,\, = \frac{1}{4}\beta \frac{{l + 2m - l^2 + 4rm + 2m^2 + 2rl}}{{\left( {r + m + 1} \right)r\left( {r + m} \right)}} > 0.} \hfill \\ \end{array} $$

Hence $s_{m}^{t+1}=0$ implies that $s_{m+1}^{t+1}=0.$□

Claim 3

An altruist cluster cannot be punctured.

Proof of Claim 3

Altruist clusters adjacent to egoist clusters of length at least 2r cannot be punctured from Lemma 1. Accordingly, let the players {1,...,l} constitute an egoist cluster with r + 1 ≤ l ≤ 2r − 1 . Consider (altruist) players l + m with m ∈ {1,...,r}. For 2r − l < m ≤ r, The average payoff of altruists is given in Eq. 5. The average payoff of egoists is given by Eq. 6. By Lemma 1, that player l + m remains an altruist. For 1 ≤ m ≤ 2r − l, the average payoff of altruists is given in Eq. 5 and, for 1 ≤ m ≤ 2r − l,

$$ \bar{\pi}_{m}^{e}=\frac{1}{r-m+1}\left( \sum_{i=1}^{l-r}\frac{\beta \left( r+1-i\right) }{2r}+\frac{\beta \left( 2r-l+1\right) \left( 2r-l+1-m\right) }{ 2r}\right) $$

Differentiating $\bar{\pi}_{m}^{a}-\bar{\pi}_{m}^{e}$ with respect to m,

$$ \frac{\partial \left( \bar{\pi}_{m}^{a}-\bar{\pi}_{m}^{e}\right) }{\partial m }=\frac{1}{4}\beta \frac{r+1}{\left( r+m\right) ^{2}}-\frac{1}{4}\beta \frac{ -2rl+r^{2}+r-l+l^{2}}{\left( r-m+1\right) ^{2}r} $$

(13)

where both $\frac{\partial \bar{\pi}_{m}^{a}}{\partial m}$ and $\frac{ \partial \bar{\pi}_{m}^{e}}{\partial m}$ are positive. Since Eq. 13 is decreasing in m,

$$ \frac{\partial \left( \bar{\pi}_{m}^{a}-\bar{\pi}_{m}^{e}\right) }{\partial m }>\frac{1}{4}\beta \frac{r+1}{\left( r+\left( 2r-l\right) \right) ^{2}}- \frac{1}{4}\beta \frac{-2rl+r^{2}+r-l+l^{2}}{\left( r-\left( 2r-l\right) +1\right) ^{2}r} $$

(14)

Note that the second term in Eq. 14 decreases in l. Substituting r + 1 for l in the second term,

$$\begin{array}{*{20}c} {\frac{{\partial \left( {\overline \pi _m^a - \overline \pi _m^e } \right)}}{{\partial m}} > \frac{1}{4}\beta \frac{{r + 1}}{{\left( {r + \left( {2r - l} \right)} \right)^2 }} - \frac{1}{4}\beta \frac{{ - 2r\left( {r + 1} \right) + r^2 + r - \left( {r + 1} \right) + \left( {r + l} \right)^2 }}{{\left( {r - \left( {2r - \left( {r + 1} \right)} \right) + 1} \right)^2 r}}} \hfill \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{4}\beta \frac{{r + 1}}{{\left( {3r - l} \right)^2 }}} \hfill \\ \end{array} $$

Noting this, suppose now that player l + m switches. For player l + m − 1, $\bar{\pi}^{a}$ decreases more than $\bar{\pi}^{e}$. Player l + m − 1 also switches. Hence the altruist cluster cannot be punctured.□

Claim 4

Two heterogeneous adjacent players cannot both switch.

Proof of Claim 4

This follows from the above results when the egoist cluster is at least 2r. Accordingly, let the players {1,...,l} constitute an egoist cluster with r + 1 ≤ l ≤ 2r − 1. We need to show that if player l switches then player l + 1 does not, and if player l + 1 switches then player l does not. Note that

$$\begin{array}{*{20}c} {\overline \pi _l^e \left( {s^t } \right) = \frac{1}{{r + 1}}\left( {\sum\limits_{i = 1}^{l - r} {\frac{{\beta \left( {r + 1 - i} \right)}}{{2r}}} + \frac{{\left( {2r - 1} \right)\beta \left( {2r + 1 - l} \right)}}{{2r}} + \frac{{\beta \left( {2r - l + 1} \right)}}{{2r}}} \right),} \hfill \\ {\overline \pi _l^a \left( {s^t } \right) = \frac{1}{r}\sum\limits_{i = 1}^r {\left( {\frac{{\beta \left( {r + i - 1} \right)}}{{2r}}} \right) - 1.} } \hfill \\ \end{array} $$

Player l remains an egoist if and only if $\bar{\pi}_{l}^{a}(s^{t})-\bar{ \pi}_{l}^{e}(s^{t})\leq 0,$ or

$$ \beta \leq \frac{4r\left( r+1\right) }{4rl-2r^{2}-5r+3l-3-l^{2}}=\beta ^{\prime \prime }\left( r,l\right) . $$

Now consider (altruist) player l + 1.

$$\begin{array}{*{20}c} {\overline \pi _{l + 1}^a \left( {s^t } \right) = \frac{1}{{r + 1}}\sum\limits_{i = 1}^{r + 1} {\left( {\frac{{\beta \left( {r + i - 1} \right)}}{{2r}}} \right) - 1} } \hfill \\ {\overline \pi _{l + 1}^e \left( {s^t } \right) = \frac{1}{r}\left( {\sum\limits_{i = 1}^{l - r} {\frac{{\beta \left( {r + 1 - i} \right)}}{{2r}}} + \frac{{\left( {2r - l} \right)\beta \left( {2r + 1 - l} \right)}}{{2r}}} \right)} \hfill \\ \end{array} $$

So this player switches if and only if $\bar{\pi}_{l+1}^{a}(s^{t})-\bar{\pi} _{l+1}^{e}(s^{t})\leq 0$ or

$$ \beta \leq \frac{4r\left( r+1\right) }{4rl-2r^{2}-5r+3l-3-l^{2}}=\beta ^{\prime \prime }\left( r,l\right) . $$

It is easily verified that $\beta ^{\prime }\left( r,l\right) <\beta ^{\prime \prime }\left( r,l\right)$ for all r and l in the admissible range. If any altruists switch (β ≤ β′) then all egoists remain egoists (β < β ^″). If any egoists switch (β > β ^″) then all altruists remain altruists (β > β′).

Claims 1–4 establish that no new clusters appear. Claim 3 establishes that, for m ≥ 2r + 1 − l, player l + m remains an altruist. Hence the egoist cluster cannot expand by more than 2r − l on either side, or 4r − 2l in all. Hence it cannot expand to exceed 4r − l ≤ 3r − 1. $\Vert$□

Lemma 6

Suppose that s ^t  ∈ L(3r,1). Then no new cluster appears.

Proof of Lemma 6

To show that no new cluster appears, we need to show that no cluster is punctured, and that two heterogeneous adjacent players cannot both switch. Given the above results, we only need to consider egoist clusters of length l where 1 ≤ l ≤ r, and their adjacent altruist clusters. Without loss of generality, let the players {1,...,l} constitute an egoist cluster 1 ≤ l ≤ r. By hypothesis, this cluster must be adjacent (on either side) to an altruist cluster of length at least 3r. Since l ≤ r all (egoist) players m ∈ {1,...,l} observe the same mean egoist payoff, given by

$$ \bar{\pi}_{m}^{e}(s^{t})=\frac{\beta \left( 2r+1-l\right) }{2r}. $$

The mean altruist payoff observed by player m is

$$\overline \pi _m^a \left( {s^t } \right) = \frac{1}{{2r + 1 - l}}\left( {\frac{{2\left( {r - l} \right)\left( {2r - l} \right)\beta }}{{2r}} + \sum\limits_{i = 1}^m {\frac{{\beta \left( {2r - l + i - 1} \right)}}{{2r}} + \sum\limits_{i = 1}^{l - m + 1} {\frac{{\beta \left( {2r - l + i - 1} \right)}}{{2r}}} } } \right) - 1.$$

This payoff $\bar{\pi}_{m}^{a}(s^{t})$ is greater for values of m closer to the boundaries of the cluster {1,...,l}. This is because players closer to the boundary are in contact with more altruists deeper within altruist clusters and hence with altruists who are earn higher payoffs. Hence player m switches to altruism only if all egoists closer to the boundary of the cluster {1,...,l} also switch. This implies that the egoist cluster cannot be punctured.

Consider whether player l remains an egoist.

$$ \bar{\pi}_{l}^{e}(s^{t})-\bar{\pi}_{l}^{a}(s^{t})=1-\frac{1}{4}\beta \frac{ -4r-2+l+l^{2}}{\left( 2r+1-l\right) r} $$

Player l (and hence all egoists) remain egoists if and only if

$$ \beta \leq 4\frac{\left( 2r+1-l\right) r}{-4r-2+l+l^{2}}=\beta ^{\prime \prime }(r,l) $$

We next show that if egoist l switches, then no altruists switch. As before, the altruist most likely to switch is player l + 1. This follows from the facts that (a) all egoists have the same payoffs and hence the mean egoist payoffs are the same for all altruists who observe an egoist, (b) all altruists in N _l + 1 are also in N _l + m where 2 ≤ m ≤ r, and (c) any altruists in N _l + m who is not in N _l + 1 is in a homogeneous neighborhood and hence obtains β − 1, the highest payoff possible for an altruist is

$$ \bar{\pi}_{l+1}^{a}(s^{t})=\frac{1}{2r+1-l}\left( \frac{2\left( r-l\right) \beta \left( 2r-l\right) }{2r}+\sum_{i=1}^{l+1}\frac{\beta \left( 2r-l+i-1\right) }{2r}\right) -1. $$

Hence

$$\begin{array}{*{20}c} {\overline \pi _{l + 1}^e \left( {s^t } \right) - \overline \pi _{l + 1}^a \left( {s^t } \right)} \hfill \\ {\,\,\,\,\,\,\, = \frac{{\beta \left( {2r + 1 - l} \right)}}{{2r}} - \frac{{\frac{{2\left( {r - l} \right)\beta \left( {2r - l} \right)}}{{2r}} + \sum\nolimits_{i = 1}^{l + 1} {\frac{{\beta \left( {2r - l + i - 1} \right)}}{{2r}}} }}{{2r + 1 - l}} + 1} \hfill \\ {\,\,\,\,\,\, = 1 - \frac{1}{4}\beta \frac{{ - 4r - 2 + 3l + l^2 }}{{\left( {2r + 1 - l} \right)r}}} \hfill \\ \end{array} $$

Player l + 1 switches if and only if

$$ \beta \leq \frac{4\left( 2r+1-l\right) r}{-4r-2+3l+l^{2}}=\beta ^{\prime }(r,l). $$

Note that β′(r,l) < β″(r,l). Hence if β > β″(r,l), which makes one or more egoists switch, then β > β′(r,l), so all altruists remain altruists. In this case no new cluster appears. On the other hand if β ≤ β′(r,l), so that one or more altruists switch, then β < β″(r,l), so all egoists remain egoists. Again no new cluster appears. Finally, if β′(r,l) < β ≤ β″(r,l), then no player in the egoist cluster and no adjacent altruist player switches. In this case too, no new cluster appears. || □

The proof of Proposition 1 may now be completed as follows. If s ⁰ ∈ L(3r,3r) then from Lemmas 1 and 2, all altruist clusters expand by 2r and all egoist clusters contract by 2r until some period t ₁ in which one or more egoist clusters are shorter than 3r. In this period, all altruist clusters are of length at least 5r, and egoist clusters fall into one of five categories; (I) length l ≥ 3r, (II) 2r + 1 ≤ l ≤ 3r − 1, (III) 2r = l, (IV) r + 1 ≤ l ≤ 2r − 1, or (V) 1 ≤ l ≤ r. From Lemmas 2–6 Clusters in categories (I) and (II) contract, those in (III) survive, contract or vanish, and those in (IV) and (V) expand, contract, survive or vanish but cannot expand to exceed length 3r. Hence no egoist cluster can return to category (I) after leaving it, and no altruist cluster can contract to a length less than 3r. Since all egoist clusters in category (I) contract in each period from Lemma 2, they must eventually fall into another category. Hence there exists τ such that for all t ≥ τ, s ^t ∈ L(3r,0) ∩ U(n,3r). Since egoist and altruist clusters alternate by definition, ρ ^t must be at least 0.5 at all states s ^t such that t ≥ τ.

Proof of Proposition 2

We start with two lemmata for any r.□

Lemma 7

Suppose that β ≤ β _h (r). Then all altruist clusters of length at least 2r contract or are punctured.

Proof of Lemma 7

Suppose β ≤ β _h (r) and let the players {1,...,l} constitute an altruist cluster, where l ≥ 2r. Consider player l + 1 − r. Note that this player has only one egoist neighbor, player l + 1. Hence $\bar{\pi}_{l+1-r}^{a}-\bar{\pi}_{l+1-r}^{e}=\bar{\pi} _{l+1-r}^{a}-\pi _{l+1}$. We claim that this difference is greatest when players {l + 2,...,l + r + 1} are all egoists. To see this, observe that if any number m of these players switches from E to A then π _l + 1 increases by βm/2r while $\bar{\pi}_{l+1-r}^{a}$ increases by strictly less than βm/2r. This is because at most r − 1 of the altruists in the set {l + 1 − 2r,...,l} have their payoffs raised, and these payoffs are raised by at most βm/2r.The remaining altruists experience no change in payoff. This proves that $\bar{\pi}_{l+1-r}^{a}-\bar{ \pi}_{l+1-r}^{e}$ is maximized when players {l + 2,...,l + r + 1} are all egoists. Hence if altruist l + 1 − r switches when players {l + 2,...,l + r + 1} are all egoists, then altruist l + 1 − r will also switch regardless of the composition of {l + 2,...,l + r + 1}. Accordingly, suppose that players {l + 2,...,l + r + 1} are all egoists. First consider the case l ≥ 3r. Then

$$ \bar{\pi}_{l+1-r}^{a}-\bar{\pi}_{l+1-r}^{e}=\frac{1}{2r}\left( \sum_{i=1}^{r} \frac{\beta \left( r+i-1\right) }{2r}+r\beta \right) -1-\frac{\beta r}{2r}= \frac{1}{8r}\left( 3r\beta -\beta -8r\right) $$

Hence if β ≤ β _h (r), altruist l + 1 − r switches, so the cluster {1,...,l} either contracts or is punctured. Next consider the case 2r ≤ l < 3r. In this case $\bar{\pi}_{l+1-r}^{e}$ is the same as it would be when l ≥ 3r, while $\bar{\pi}_{l+1-r}^{a}$ is strictly less than it would be when l ≥ 3r. Since altruist l + 1 − r switches when l ≥ 3r, this player must also switch when 2r ≤ l < 3r. ||□

Lemma 8

Suppose that β ≤ β _h (r). Then any egoist who have only one altruist in his neighborhood remain as egoist.

Proof of Lemma 8

Suppose β ≤ β _h (r). The egoist we consider must belong to the egoist cluster of length at least 2r. Let the players {1,...,l} constitute an egoist cluster, where l ≥ 2r. Consider player l + 1 − r. Note that this player has only one altruist neighbor, player l + 1. Hence $\bar{\pi}_{l+1-r}^{a}-\bar{\pi} _{l+1-r}^{e}=\pi _{l+1}-\bar{\pi}_{l+1-r}^{e}$. We claim that this difference is greatest when players {l + 2,...,l + r + 1} are all altruists. To see this, observe that if any number m of these players switches from A to E then π _l + 1 decreases by βm/2r while $\bar{\pi} _{l+1-r}^{a}$ decreases by strictly less than βm/2r. This is because at most r − 1 of the egoists in the set {l + 1 − 2r,...,l} have their payoffs decreased, and these payoffs are decreased by at most βm/2r. The remaining egoists experience no change in payoff. This proves that $\pi _{l+1}-\bar{\pi}_{l+1-r}^{e}$ is maximized when players {l + 2,...,l + r + 1} are all altruists. Hence if egoist l + 1 − r remain an egoist when players {l + 2,...,l + r + 1} are all altruists, then egoist l + 1 − r will also remain an egoist regardless of the composition of {l + 2,...,l + r + 1}. Accordingly, suppose that players {l + 2,...,l + r + 1} are all altruists. First consider the case l ≥ 3r. Then

$$ \bar{\pi}_{l+1-r}^{a}-\bar{\pi}_{l+1-r}^{e}=\frac{\beta r}{2r}-1-\frac{1}{2r} \left( \sum_{i=1}^{r}\frac{\beta i}{2r}\right) =\frac{1}{8r}\left( 3r\beta -\beta -8r\right) $$

Hence if β ≤ β _h (r), egoist l + 1 − r remains an egoist. Next consider the case 2r ≤ l < 3r. In this case $\bar{\pi}_{l+1-r}^{e}$ is strictly greater than it would be when l ≥ 3r, while $\bar{\pi} _{l+1-r}^{a}$ is the same as it would be when l ≥ 3r. Since egoist l + 1 − r remains when l ≥ 3r, this player must also remain an egoist when 2r ≤ l < 3r. ||

Now assume that r = 2 and thus $\beta _{h}\left( 2\right) =\frac{16}{5}$ Let $A\left( m\right)$ denote the altruist cluster of length m and $E\left( m\right)$ for the egoist cluster of length m. Let a and e with superscript ∗ is the player in $A\left( m\right)$ and $E\left( m\right)$ respectively, while $a\left( e\right)$ be any other altruist(egoist). Let x denote the player with the unspecified type. Lemma 8 implies that with r = 2, the egoist cluster is never punctured. This and Lemma 7 implies that any newly formed altruist cluster will be of length at most 2.□

Claim 1

All the altruist in $A\left( m\right)$ for m ≤ 4 switch to egoist.

Proof of Claim 1

Note that we can possibly consider two type of altruists in $A\left( m\right)$: either (Case I) all the egoist in his neighborhood has only one altruist neighbor, namely the altruist in $A\left( m\right)$, or (Case II) at least one egoist in the neighborhood whose payoff is at least $\frac{2}{4}\beta$. Case I is feasible iff e,e,e,e,a∗,e,e,e,e. where it is easy to see that this altruist will switch to egoist next period( this is true for any β). Case I implies that if not Case I , all the altruists in $A\left( m\right)$ have at least one egoist neighbor whose payoff is at least $\frac{2}{4}\beta$.

Now consider Case II. There are two possible cases: either (Case II-1) at least one altruist in his neighborhood has the payoff of less than $-1+\frac{3}{4}\beta$ or (Case II-2) all altruist neighbors including himself have the payoff $-1+\frac{3}{4}\beta$. In Case II-2, all the egoist the altruists face have the payoff $\frac{3}{4}\beta .$ Thus these altruists will convert to egoist. Consider Case II-1. There are two possible cases for this: either there is only one egoist neighbor for $A\left( m\right)$ who has at least two altruist neighbors or there are more than one such egoist neighbor. The first is only feasible with $x_{1},x_{2},a,e,a^{\ast },e,e,e,e$ (and its mirror $e,e,e,e,a^{\ast },e,a,x_{2},x_{1}$). The best case for the altruist a∗ to remain an altruist is when x ₁ = x ₂ = a. In this case

$$ \bar{\pi}^{a}-\bar{\pi}^{e}= \frac{1}{12}\beta -1\leq -\frac{11}{ 15}<0 $$

Hence this altruist player will switch to egoist. For the second case, the maximum of $\bar{\pi}^{a}$ among $A\left( m\right)$ is

$$ \bar{\pi}^{a}=\frac{\frac{2}{4}\beta +\left( n-1\right) \frac{3}{4}\beta }{n} -1\text{ \ for }n=1,2,3 $$

while the minimum of the average payoff of egoist around the altruist in $A\left( m\right)$ is

$$ \bar{\pi}^{e}=\frac{2\frac{2}{4}\beta +\left( 5-n-2\right) \frac{1}{4}\beta }{5-n}\text{ \ for }n=1,2,3 $$

The difference is

$$ \bar{\pi}^{a}-\bar{\pi}^{e}=\frac{1}{4}\frac{5\beta -9\beta n+2\beta n^{2}+20n-4n^{2}}{n\left( -5+n\right) } $$

which is increasing in β for all n. Substituting $\frac{16}{5}$ for β and then we have

$$\bar{\pi}^{a}-\bar{\pi}^{e}=-\frac{1}{5}\frac{20-11n+3n^{2}}{n\left(5-n\right) }<0$$

Claim 2

All the egoist in $E\left( m\right)$ for m ≤ 4 will remain an egoist.

Proof of Claim 2

All the egoist neighbors of the egoist in $E\left( m\right)$ including himself is of payoff at least $\frac{1}{4}\beta$. There are possibly two types of egoists: either (Case I) all the egoist in the neighborhood including himself face only one altruist. or (Case II) at least one egoist in the neighborhood including himself has the payoff of at least $\frac{2}{4}\beta$. Case I is feasible iff the altruist neighbor is surrounded by egoists and thus the average payoff of this altruist is $-1+\frac{1}{4}\beta$. Thus $\bar{\pi}^{a}<\bar{ \pi}^{e}$ for the egoists in $E\left( m\right) .$ Now consider Case II. Notice that amongst the altruist neighbors that the egoist in $E\left( m\right)$ face, there exist at least one altruist neighbor whose payoff is less than $-1+\frac{3}{4}\beta$, except the case of

$$ a,a,a,a,e^{\ast },a,a,a,a $$

in which case the egoist will survive as egoist. Other than this case, we can possibly consider two separate cases: either (Case II-1) there is only one altruist whose payoff is at most $-1+\frac{2}{4}\beta$ in the neighborhood of $E\left( m\right)$ or (Case II-2) there are more than one such altruist. Case II-1 is feasible only with $x_{1},x_{2},e,a,e^{\ast },a,a,a,a$ (and its mirror image) and e,a,a,a,e∗,a,a,a,a (and its mirror image). It is easy to see that e∗ will remain in the latter case. For the first case, the best situation for the egoist e∗ to switch to altruist is x ₁ = x ₂ = e. In this case, $\bar{\pi}^{e}=\frac{1}{2}\left( \frac{3}{4}\beta +\frac{1}{4 }\beta \right) =\frac{1}{2}\beta$ and $\bar{\pi}^{a}=\frac{1}{3}\left( \frac{6}{4}\beta +\frac{1}{4}\beta \right) -1= \frac{7}{12}\beta -1$. Thus $\bar{\pi}^{a}-\bar{\pi}^{e}= \frac{1}{12}\beta -1< -\frac{11}{15}<0$. Hence the egoist will remain as egoist. Now consider Case II-2. The minimum average payoff of egoists in $E\left( m\right)$ is

$$ \bar{\pi}^{e}=\frac{\frac{2}{4}\beta +\left( 5-n-1\right) \frac{1}{4}\beta }{ 5-n}\text{ \ \ for }n=2,3,4 $$

while the maximum average payoff of altruists in $E\left( m\right)$ is

$$ \bar{\pi}^{a}=\frac{2\frac{2}{4}\beta +\left( n-2\right) \frac{3}{4}\beta }{n }-1\text{ \ for }n=2,3,4 $$

Hence

$$ \bar{\pi}^{a}-\bar{\pi}^{e}=-\frac{1}{4}\frac{10\beta -11\beta n+2\beta n^{2}+20n-4n^{2}}{n\left( 5-n\right) } $$

which is increasing in β. Substituting $\frac{16}{5}$ for β, $\bar{\pi}^{a}-\bar{\pi}^{e}=-\frac{1}{5}\frac{40-19n+3n^{2}}{n\left( 5-n\right) }<0$.

What remains to be shown is that the newly created altruist cluster after punctured by egoists will eventually disappear without expanding indefinitely into the egoist cluster. We claim that this newly created altruist cluster will disappear in the very next period without seeding altruist in the egoist cluster.□

Claim 3

The altruist in $A\left( m\right)$ for m = 1,2 disappear and do not expand.

Proof of Claim 3

For m = 1,we have the following situation; $...e_{2},a^{\ast },e_{1},...$.By Claim 1, a ∗ will switch to egoist. We need to show that the surrounding egoists $e_{1}\left( e_{2}\right)$ will remain as egoist in the next period. This is sufficient since the egoist cluster can not be punctured by Lemma 8 for r = 2. We will consider e ₁ only since the result also applies to e ₂ by symmetry. By Claim 2, it is sufficient to consider the case of $x_{1},x_{2},e_{2},a^{\ast },e_{1},E\left( 4\right)$. It is easy to see that the worst situation for e ₁ to remain an egoist is x ₁ = e and x ₂ = a. In this case, $\bar{ \pi}_{e_{1}}^{a}-\bar{\pi}_{e_{1}}^{e}=-1<0$ Thus e ₁will remain an egoist.

Suppose now m = 2. Following argument for m = 1, it is sufficient to consider the case of $x,e_{2},a^{\ast },a^{\ast },e_{1},E\left( 4\right)$. The worst situation for e ₁ to remain an egoist is x = a. Then

$$\begin{array}{*{20}c} {\overline \pi _{e_{_1 } }^e = \frac{{\frac{2}{4}\beta + \frac{1}{4}\beta }}{3} = \frac{1}{4}\beta } \hfill \\ {\overline \pi _{e_{_1 } }^a = \frac{{\frac{2}{4}\beta + \frac{1}{4}\beta }}{2} - 1 = \frac{3}{8}\beta - 1} \hfill \\ \end{array} $$

Hence $\bar{\pi}_{e_{1}}^{a}-\bar{\pi}_{e_{1}}^{e}=$ $\frac{1}{8}\beta -1$. Substituting $\frac{16}{5}$ for β, $\bar{\pi}_{e_{1}}^{a}-\bar{\pi} _{e_{1}}^{e}<-\frac{3}{5}<0$. Thus e ₁ will remain as egoist.

The proof of Proposition 2 can be completed as follows. By Lemma 7, the altruist cluster of length at least 2r either contracts until its length is less than or equal to 2r and then it disappears by Claim 1 and 3 or it is punctured and the newly created altruist cluster, whose length is at most 2r by Lemma 8, disappears without expansion by Claim 1 and 3. Eventually the length of the cluster reaches at most 2r, at the point in which it disappears.□