Dynamic Conflict on A Network∗†
Marcin Dziubiński‡
Sanjeev Goyal§
David E. N. Minarsch¶
May, 2016
Abstract
Players are endowed with resources. A player can engage in conflict with others
to enlarge his resources. The set of potential conflicts is defined by a contiguity
network. Players are farsighted and aim to maximize their resources. They decide
on whether to wage war or remain peaceful. The winner of a war takes control
of the loser’s node and resources; he then decides on whether to wage war against
other neighbours, or to stay peaceful. The game ends when either all players choose
to be peaceful or when only one player is left.
We study the influence of the technology of conflict and the structure of the
network on war and peace. Our first result identifies a threshold property in the
technology of conflict: above this threshold, every ruler wishes to wage war and,
eventually, there is a hegemon. Below the threshold, resources and networks determine the prospects for peace and the number of kingdoms. We develop sufficient
conditions for war and provide examples with lasting peace.
1
Introduction
Individuals seek to accumulate more resources. One possible avenue to obtain additional
resources is to appropriate them through conflict. There is wide ranging evidence that
conflict usually takes place among physically contiguous entities Caselli et al. (2014).
When Mr A wins in a fight with Mr B this brings him in contact – and potentially in
conflict – with those who are close to Mr B. When Mr A fights with Mr B, the technology
of conflict will map their relative resources onto the respective probabilities of winning.
Thus the technology of conflict, the resources of Mr B, his connections, and the resources
of his neighbours will all matter in Mr A’s decision on whether to wage a war on Mr B
– now or later – or to remain peaceful. In this paper, our aim is develop a framework
that will help in understanding the dynamics of conflict and resource accumulation, in
structured populations.
We consider a set of players who each control sets of nodes of a network whose links
represent possible conflict pairs. Each node is allocated (non-transferable) resources,
∗
This is full version of the paper accepted for the EC’16: 17th ACM Conference on Economics and
Computation, Maastricht 2016.
†
Marcin Dziubiński acknowledges support from Polish National Science Centre through grant nr
2014/13/B/ST6/01807. Sanjeev Goyal acknowledges support from a Senior Keynes Fellowship. Sanjeev
Goyal and David Minarsch acknowledge support from the Cambridge-INET Institute.
‡
Institute of Informatics, University of Warsaw. E-mail: [email protected]
§
Faculty of Economics & Christ’s College, University of Cambridge. E-mail: [email protected]
¶
Faculty of Economics & Girton College, University of Cambridge. E-mail: [email protected]
1
which are controlled by the player controlling that node. A player can appropriate
resources from other players by engaging in a conflict with them. The outcome of a
bilateral conflict depends on the resources controlled by the conflicting players and is
given by the Tullock contest function.1 A player chooses between two courses of action:
remaining peaceful (not entering into a conflict with any neighbour) and engaging in
a sequence of conflicts with specified neighbours (and potentially the neighbours of
neighbours, and so forth).
The game proceeds in rounds. At the beginning of each round, a player is picked at
random and implements her strategy. If the attacker loses, the round ends. Otherwise,
the attacker is allowed to attack neighbours until she loses, or chooses to stop, or there
are no neighbours to attack. After a conflict between two players, the winner takes over
the nodes and the resources controlled by the loser. The game continues until either no
player wishes to fight (peace) or only one player is left (hegemony).
We identify a threshold property of conflict: independent of the network and resource
configuration, a peace equilibrium cannot be sustained with two or more players if
the technology of conflict favours the larger resource contestant – when the Tullock
parameter γ > 1. This result is driven by two features of the contest success function:
when γ > 1 a conflict yields higher expected resources to the stronger player and it
is always better to attack than to wait for other players to have engaged in conflict
with each other. On the other hand, if technology of conflict favours the weaker player
– i.e., if the Tullock parameter γ < 1, then it is always better to wait and let other
players fight. We develop sufficient conditions on resources and network configurations
for perpetual strife or lasting peace.
Our paper contributes to the study of conflict and to the study of networks.
In the context of the literature on conflict, the innovation we offer is a model of
the dynamics of inter-connected conflict. For a survey of the literature on conflict see
Garfinkel and Skaperdas (2012) and Konrad (2009). To the best of our knowledge,
the work on dynamic incentives for conflict restricts attention to two player problems.
Garfinkel and Skaperdas (2000) analyse a two-period model where in the first stage of
each period investment in conflict and production effort choices are made simultaneously
by both players and in the second stage players decide whether to enter a contest or
not. They show that players have an incentive to engage in conflict if the future matters
sufficiently and conflict is not too costly. In a recent paper, McBride and Skaperdas
(2014) show that conflict today may act as an investment in the future strength of a
player. Our analysis explores the scope of this idea within a general framework with
interconnected individuals. We find that the incentive for fighting always prevails if the
technology of conflict is above a threshold, but not otherwise. In the latter situation,
the configuration of resources and networks both matter and may make it optimal to
postpone fighting or not to fight at all. The analysis yields us insights into the circumstances when we may expect to see the emergence of a single hegemon and when
multiple entities can live in peace.
In the context of the literature on network, our contribution is a dynamic model of
resource accumulation in a network. For a general survey of research on networks see
Goyal (2016) and for a survey of the research on networks and conflict see Dziubiński
et al. (2016). In a recent paper, Franke and Öztürk (2015) analyse how conflict is shaped
by network structure. They study a one-shot conflict on a given network, where each
1
If Mr A devotes x and Mr B devotes y then the probability of Mr A winning is xγ /(xγ + y γ ), where
γ > 0, is reflects the importance of resources and is referred to as the technology of conflict (Hirshleifer
(1995)).
2
player chooses investment in conflict on each of her links. The main result characterizes
investments on classes of networks. In a similar spirit, König et al. (2014) study a oneshot model of conflict in a network where links may be positive (as between allies) or
negative (as between enemies). The authors restrict attention to a linear Tullock contest
function (γ = 1). Investments by allies are strategic substitutes, while investments
by enemies are strategic complements. Equilibrium investments are proportional to
Bonacich centrality of the allies and enemies networks, respectively. By contrast, the
focus of our work is on the dynamics of conflict across interconnected entities. Our
analysis highlights the key role of the technology of conflict, the distribution of resources
and the network of contiguity in determining whether a single hegemon prevails or if
multiple entities exist in a lasting peace.
Our paper builds on De Jong et al. (2014): they have a similar resource and network
formulation, but conflict is imposed exogenously. Links are picked at random and player
must fight. By contrast, in the present paper, waging a war or being at peace are choice
variables. This difference leads to a very different analysis and also to very different
results: in De Jong et al. (2014) there is always a hegemon, at the end. By contrast,
a key result in our paper pertains to the circumstances that lead to a lasting peace,
with multiple kingdoms. Our analysis also develops general results on the role of the
technology of conflict that were beyond the scope of the earlier paper.
The rest of the paper is organized as follows. Section 2 presents the model. Section 3
presents the results. We conclude in Section 4. Some of the proofs are moved to the
Appendix.
2
The model
Let N = {1, 2, . . . , n} be a set of n ≥ 2 players and V = {1, . . . , n} be a set of vertices.
Every vertex v ∈ V is endowed with a positive amount of resources, rv ∈ R++ . Every
vertex v ∈ V is owned by exactly one player, o(v) ∈ N , where o : V → N is called
the ownership function. Owning a node, a player owns resources the node is endowed
with. The resources of player i ∈ N under ownership function o, Ri (o), is the sum of
resources allocated to the nodes owned by the player:
Ri (o) =
X
v∈o
rv
(1)
−1 (i)
The vertices are connected in a network, represented by an undirected graph G =
hV, Ei, where E = {uv : u, v ∈ V, u 6= v} is the set of edges (or links) in G. The
network together with the ownership function induce a neighbour relation between the
players: two players i, j ∈ N are neighbours in network G = hV, Ei (under ownership
function o) if there exists u ∈ V , owned by i, and v ∈ V , owned by j, such that uv ∈
E. Figure 1 illustrates resources endowment, ownership function and neighbourhood
relation between players in an exemplary network.
Players who are neighbours can engage in a conflict. The probability that player
i, with resources Ri , wins the conflict with player j, with resources Rj , is determined
by a contest success function (CSF) p : R2++ → [0, 1]: player i wins with probability
p(Ri , Rj ) (Tullock (1980)), where
p(x, y) =
3
xγ
,
xγ + y γ
(2)
Figure 1: (a) A network with resource endowment indicated within nodes and ownership function indicated by thick lines encircling nodes owned by the same player; (b)
Neighbourhood relation between players induced by the network and the ownership
function.
where γ ∈ R+ ∪ {+∞} captures the marginal increase in the probability of winning
caused by increasing the resources.2 The function is widely used in the study of conflict
and has a number of desirable axiomatic properties (c.f. Skaperdas (1996)).
We study a dynamic game between the players. There are rounds numbered t =
1, 2, . . .. At the start of a round, one of the players (say) i is picked at random. He
chooses either to be peaceful or to attack one of his neighbours. Player i wins with
probability p(Ri , Rj ), and player j wins otherwise. If the attacker loses, the round ends.
Otherwise, the attacker is allowed to attack neighbours until she loses, or chooses to
stop, or there are no neighbours to attack. Winning a conflict, the attacker takes over
the vertices of the losing player (and their connections), together with the resources
owned by them. As he takes over the connections of the newly acquired nodes, the set
of neighbours also changes. The game ends when all players choose to be peaceful (the
case of a single surviving player is a special case). The objective of each player is to
maximise her expected resources at the end of the game.
We now describe the dynamics more formally. Each state of the game is determined
by an ownership function, and the set of states of the game is3
O(G) = {o ∈ N V : for all i ∈ N , G[o−1 (i)] is connected}.
(3)
At every state of the game, o ∈ O(G), each player i ∈ N chooses a sequence of players,
si (o), to attack. A sequence of players, σ, is feasible for player i at state o in G if it is
either empty, σ = ε, or σ = j1 , . . . , jk and for all 1 ≤ l < k, jl ∈
/ {i, j1 , . . . , jl−1 } and
jl is a neighbour of one of the players from {i, j1 , . . . , jl−1 } in G under o. A strategy
of player i is a function si : O(G) → N ∗ such that for each state o ∈ O(G), si (o) is
feasible.4 Notice that the only feasible sequence for players who do not own any vertices
at the given state, as well as for the player who owns all the vertices at the given state,
is the empty sequence. A non-empty feasible sequence is called an attacking sequence.
We now turn to defining the expected payoffs of the players. To this end we need
to introduce some basic preliminary concepts. Players in the game engage in sequences
2
The case of γ = +∞ is all-pay contest where the stronger player wins with probability 1.
Given a set of vertices U ⊆ V , G[U ] = (U, {vu ∈ E : v, u ∈ U }) is the subgraph of G induced by U ,
i.e. the subgraph of G restricted to vertices in U and links between them.
4
Given set N , we use the standard notation N ∗ to denote the set of finite sequences over N , and ε
to denote the empty sequence.
3
4
of conflicts where the player winning subsequent conflicts accumulates the resources
of the opponent. To express the probability of winning such sequences of conflicts we
introduce the following notation. Given a sequence x1 , x2 , . . . , xm of at least two positive
real numbers, xi ∈ R++ for i ∈ {1, . . . , m}, let


m
k−1
Y
X
pseq (x1 , . . . , xm ) =
p
xj , xk  ,
(4)
k=2
j=1
be the probability that a player with resources x1 wins a sequence of conflicts with
players with resources x2 , . . . , xm , accumulating the resources of the loosing opponents
at each step of the sequence. For convenience we also assume that pseq (x) = 1.
To express the state that results from a player executing successfully a given number
attacks from the chosen sequence, we introduce the following notation. Given player i,
state o ∈ O(G), a sequence of attacks, a, of i and number k ∈ {0, . . . , m}, the resulting
state in the case of i winning k conflicts on the sequence and, in the case of k < m,
loosing the k + 1’st one, is denoted by o[k|i; a]. Given v ∈ V ,
o[0|i; ε](v) = o(v)

if k = m and o(v) ∈ {j1 , . . . , jk },
 i,
o[k|i; j1 , . . . , jm ](v) = jk+1 , if 0 ≤ k < m and o(v) ∈ {j1 , . . . , jk+1 },

o(v), otherwise.
Given state
o and strategy profile s = (s1 , . . . , sn ), let
Atck(s, o) = {i ∈ N : si (o) 6= ε}
(5)
be the set of attacking players at o under s. The strategy profile s and an initial
state o determine a probability distribution on the final states of the game. So, for
sj0 (o) = (j1 , . . . , jm ), the expected payoff to player i at state o under strategy profile s
when an attacking player j0 ∈ Atck(s, o) is selected to move is
Π(i, o, s|j0 ) = pseq (Rj0 (o), . . . , Rjm (o))Π (i, o[m|j0 ; j1 , . . . , jm ]|s) +
!
m−1
k
X
X
pseq (Rj0 (o), . . . , Rjk (o))p Rjk+1 (o),
Rjk (o) Π (i, o[k|j0 ; j1 , . . . , jm ], s) . (6)
k=0
l=0
We can now define the expected payoff to player i at state o under strategy profile
s as
(
R
if Atck(s, o) = ∅,
P
i (o),
(7)
Π(i, o, s) =
1
j∈Atck(s,o) Π(i, o, s|j), otherwise,
|Atck(s,o)|
Every player aims to maximise her expected payoff. We study the Markov perfect
equilibria of the game (called equilibria in the remaining part of the paper). Thus we
assume that choices of the players are independent of the history and are time consistent.
The game is finite (after every attack the number of active players falls) and sequential. Hence standard results guarantee existence of (Markov perfect) equilibrium.
Proposition 2.1. For any graph G, resource vector r ∈ RV++ , and any state
the game has a Markov perfect equilibrium.
5
o ∈ O(G),
3
Analysis
In the analysis we focus on two distinct cases of the Tullock CSFs: γ > 1 and γ < 1.
Suppose x > y. The following inequalities play an important role in our analysis.
1. If γ > 1 then (x + y)p(x, y) > x (strong rewarding).
2. If γ < 1, then (x + y)p(x, y) < x (weak rewarding).
Under a strong rewarding CSF, the expected resources of the stronger player are
higher than her current resources and the expected resources of the weaker player are
lower. In the case of a weak rewarding CSF, this is reversed. It is worth noting that
if the players are myopic, no peace is possible, regardless of the network and resource
allocation (barring the non-generic case of γ = 1). Given any two players with unequal
resources, one of them will increase his expected resources by engaging in conflict.
During the course of the game subsequent conflicts between pairs of players dispossess
the loosing players of their vertices and in effect leave them with no choice of strategy:
they can only choose the empty sequence. Similarly, the player who owns all vertices
has no choice but to choose the empty sequence. We call such players inactive. The
remaining players are called active: these are exactly the players who own at least one
but not all vertices. Given state o, the set of active players at o is
Act(o) = {i ∈ N : ∅ ( o−1 (i) ( V }.
(8)
We are interested in the equilibrium choices of the active players.
3.1
Strong rewarding conflict
We show in this section that when γ > 1, then for any network and any resources
endowment, any equilibrium leads to a hegemony: there is exactly one player owning all
vertices. Hence it is never possible to have peace in this case. This result is driven by
the fact that Tullock CSF is strong rewarding. A player who has a feasible sequence of
attacks such that she is always the stronger contestant is never willing to choose peace
when it is chosen by all other players. In addition to that, in any equilibrium (barring a
case of |V | = 2), every player chooses a sequence of attacks involving all the remaining
active players or, in other words, chooses to attack till hegemony. An even stronger
property holds: at every state there is a non-empty set of players for whom choosing
attack till hegemony is a dominant strategy. This is driven by another feature of the
Tullock CSF with γ > 1, stated in the lemma below (proven in the Appendix).
Lemma 3.1. Let p : R2++ → [0, 1] be a Tullock CSF with γ > 1. Then for all x, y, z ∈
R++ ,
p(x, y)p(x + y, z) > p(x, y + z).
(9)
Property (9) implies that attacking opponents in a sequence gives higher chance of
winning than waiting for them to merge and attack them afterwards. This is stated in
the corollary below.
Corollary 3.2. Let m ≥ 3, x1 , . . . , xm ∈ R++ , and 1 ≤ i < j ≤ m such that i 6= 1 or
j 6= m. Then
!
j
X
pseq (x1 , . . . , xi−1 , xi , . . . , xj , xj+1 , . . . , xm ) > pseq x1 , . . . , xi−1 ,
xl , xj+1 , . . . , xm
l=i
(10)
6
As we already mentioned, sequences of players to attack are particularly interesting
choices of the active players when CSF is strong rewarding. We define such attacking
sequences more formally. Let o be a state and i ∈ Act(o) be an active player at o. A
permutation of set Act(o) \ {i}, τ , such that the sequence σ = i, τ is feasible for i in G
under o is called a full attacking sequence. After executing a full attacking
Psuccessfully,
player i controls all the nodes and all the resources in the network, R = v∈V rv . and
becomes a hegemon. A full attacking sequence, σ, that maximises the expected payoff
of player i or, equivalently, maximises the probability pseq (σ), of being successfully
executed, is called an optimal full attacking sequence. A particular type of attacking
sequences are those where the attacking player has higher resources than the opponent
for every conflict along the sequence. We call such a sequence a strong attacking sequence.
Formally, an attacking sequencePσ = i1 , . . . , ik of attacking player i0 is strong at a given
state o if for all l ∈ {1, . . . , k}, l−1
j=0 Rij (o) > Ril (o). The set
Str(G, o) = {i ∈ Act(o) : there exists a strong f.a.s. σ for i at o}
(11)
is the set of strong players in network G at state o.
Conditioned on being picked to make a choice at state o, strong players have a strategy increasing their expected resources. As we show below, regardless of the strategies
chosen by the other players, if a player is strong at a given state, then choosing an
optimal full attacking sequence is her dominant choice at that state. Moreover, at every
state with active players, there always exists a strong player. Because of that, every
equilibrium leads to hegemony. This, together with the fact that choosing attack is
always better than waiting for others to fight, makes an optimal attacking sequence a
best choice for the weak players as well.
The fact that it is always better to attack than to wait has implications on the
dynamics of the set of strong players. When other players move, the expected payoff
from available optimal full attacking sequences (weakly) falls. Hence a player may cease
being strong and the set of strong players shrinks.
Theorem 3.3. Suppose γ > 1. For any state
any resource vector r ∈ RV++ :
o ∈ O(G), any connected network G, and
1. Generically, every equilibrium outcome contains a hegemon.
2. If |Act(o)| ≥ 3 then in equilibrium every active player chooses an optimal full
attacking sequence.
3. At any state o with Act(o) 6= ∅, there exists a strong player, and for any strong
player i ∈ Str(G, o), an optimal full attacking sequence is a best choice, regardless
of the strategies chosen by the other players.
Proof. Fix a connected graph G and a resource endowment r ∈ RV++ . The proof proceeds
in three steps.
Step 1: For any o with |Act(o)| ≥ 2 there exists a strong player Let o be a
state with |Act(o)| = m ≥ 2. Take any active player j0 ∈ Act(o) with maximal amount
of resources Rj0 (o). Pick any sequence j1 , . . . , jm−1 of players in Act(o) \ {j0 } that is
feasible for j0 in G under o (clearly such a sequence exists because G is connected). Since
7
j0 has maximal amount of resources so, for all 1 ≤ k ≤ m − 1 we have
Rjk (o). The expected payoff to player j0 from the attacking sequence is
! m−1
!
m−1
k−1
X
Y
X
Π(j0 , o; j1 , . . . , jm−1 ) =
Rjl (o)
p
Rjl (o), Rjk (o)
l=0
= Rj0 (o)
m−1
Y
k=1
p
k=1
k−1
X
Pk−1
l=0
Rjl (o) ≥
l=0
Rjl (o), Rjk (o)
!
l=0
Pk
o)
.
o)
l=0 Rjl (
Pk−1
l=0 Rjl (
!
(12)
Since p is strong rewarding, so
p
k−1
X
!
Rjl (o), Rjk (o)
l=0 Rjl (o)
Pk
Pk−1
l=0
l=0
!
Rjl (o)
≥ 1,
(13)
with equality only if k = 1 and Rj0 (o) = Rj1 (o). Thus Π(j0 , o; j1 , . . . , jm−1 ) ≥ Rj+0 (o)
and so j0 ∈ Str(G, o).
Since in every state with at least three players there exists a strong player whose
optimal full attacking sequence yields strictly higher expected resources, so in any equilibrium and in any such state there is at least one player who chooses attack. Thus in
any equilibrium outcome there are at most two active players. Moreover, generically, at
every state with two active players one of the players own strictly more resources than
the other player. Consequently, in any equilibrium that player chooses attack at such
a state. Hence generically there is a hegemon in any equilibrium outcome. This shows
point 1 of the theorem.
Step 2: For any player i ∈ N , any strategy profile s, any state
any feasible sequence σ of i at o maximising Π(i, o; σ),
Π(i, o; σ) ≥ Π(i, o, s).
o ∈ O(G) and
(14)
Take any player i ∈ N and any strategy profile s. The proof is by induction on the
number of active players.
For the induction basis assume that |Act(o)| = 2. Let j be the other active player.
An optimal feasible sequence of i at o is either σ = j (if Ri ≥ Rj ) or σ = ε, otherwise.
Since CSF is strong rewarding, σ yields payoff at least as high as the payoff from s at
o.
For the induction step suppose that |Act(o)| ≥ 3, i.e. there are at least three active
players, and suppose that the induction hypothesis holds if the number of active players
is less than |Act(o)|. Let σ be a feasible sequence of i at o that maximises Π(i, o; σ).
It there are no attacking players at o under strategy profile s, Atck(s, o) = ∅,
then (14) holds because, by optimality of σ,
Π(i, o; σ) ≥ Π(i, o; ε) = Π(i, o, s).
(15)
For the remaining part of the proof assume that Atck(s, o) 6= ∅.
We show first, for any j0 ∈ Atck(s, o), that
Π(i, o; σ) ≥ Π(i, o, s|j0 ).
(16)
We start with the case of j0 6= i. Take any j0 ∈ Atck(s, o) \ {i}. Let sj0 (o) =
j1 , . . . , jm be the attacking sequence of j0 at o and let o0 = o[k|j0 ; j1 , . . . , jm ] be a state
8
reached after j0 wins exactly 0 ≤ k ≤ m attacks in the sequence. Let jl be the winner
of the sequence of conflicts (where l = k + 1, if k < m, and l = 0, if k = m).
Suppose that i is not attacked when the sequence of attacks is executed, i.e. i ∈
/
{jl , j1 , . . . , jk }. We will show that in this case, for any optimal attacking sequence of i
at o, σ,
Π(i, o; σ) > Π(i, o0 , s).
(17)
Let σ 0 be an optimal feasible sequence of i at o0 . Since |Act(o0 )| < |Act(o)| so, by the
induction hypothesis,
Π(i, o0 ; σ 0 ) ≥ Π(i, o0 , s).
(18)
If σ 0 does not contain jl then it is a feasible sequence of i at o. Optimality of σ together
with (18) yield
Π(i, o; σ) ≥ Π(i, o; σ 0 ) = Π(i, o0 ; σ 0 ) ≥ Π(i, o0 , s).
(19)
If σ 0 contains player jl then σ 0 = σ10 , jl , σ20 , where σ10 and σ20 are the subsequences of
σ 0 preceding and following jl , respectively. Let τ be a permutation of {j1 , . . . , jk , jl }
such that σ 00 = σ10 , τ, σ20 is a feasible sequence for i at o (obviously such a permutation
exists). In the event of winning all the conflicts in sequence σ 0 , player i owns the same
nodes and resources as in the event of winning allPthe conflicts in the sequence σ 00 . The
resources owned by jl at o0 , Rjl (o0 ) = Rjl (o) + kq=1 Rjk (o) so, by Corollary 3.2, the
probability of player i winning all the conflicts in σ 00 is higher than the probability of
i winning all the conflicts in σ 0 . Hence Π(i, o; σ 00 ) > Π(i, o0 ; σ 0 ). By optimality of σ,
Π(i, o; σ) ≥ Π(i, o; σ 00 ) > Π(i, o0 ; σ 0 ) and, by (18),
Π(i, o; σ) > Π(i, o0 , s).
(20)
Suppose that i is attacked when the sequence leading to o0 is executed, i.e. i ∈
{jl , j1 , . . . , jk }. If i is not the winner of the sequence, i 6= jl , then payoff to i at o0 is 0.
Suppose that i is the winner of the sequence, i = jl . Since |Act(o0 )| < |Act(o)| so, by
the induction hypothesis, for any feasible sequence of i at o0 , σ 0 ,
Π(i, o0 ; σ 0 ) ≥ Π(i, o0 , s).
(21)
Let τ be a permutation of {j0 , . . . , jl−1 } such that the sequence of attacks σ 00 = τ, σ 0 is
feasible for i at o. In the event of winning all the conflicts in sequence σ 0 , as well as in
the event of winning all the conflicts in the sequence σ 00 , player i owns the same nodes
and the same resources in the network. By Corollary 3.2, the probability of player i
00 is higher than the probability of i winning the conflict
winning all the conflicts in σP
0
with player j0 with resources q−1
l=0 Rl (o) and then winning all the conflicts in σ . Hence
Π(i, o; σ 00 ) > p Rjq (o),
q−1
X
!
Rl (o) Π(i, o0 ; σ 0 ).
(22)
l=0
By optimality of σ, Π(i, o; σ) ≥ Π(i, o; σ 00 ) and, by (20) and (21),
Π(i, o; σ) > p Rjq (o),
q−1
X
!
Rl (o) Π(i, o0 , s).
(23)
l=0
If i is not on the attacking sequence of j0 at o, i ∈
/ {j1 , . . . , jm } then, by (6)
and (19), for any feasible sequence of i at o, σ, maximising Π(i, o; σ) we have Π(i, o; σ) ≥
9
Π(i, o, s|j0 ). On the other hand, if i ∈ {j1 , . . . , jm } then (6) can be rewritten as follows:
Π(i, o, s|j0 ) =
pseq (Rj0 (o), . . . , Rjq−1 (o))p Rjq (o),
q−1
X
!
Rjl (o) Π (i, o[q − 1|j0 ; j1 , . . . , jm ], s) +
l=0
q−1
X
k=0
pseq (Rj0 (o), . . . , Rjk (o))p Rjk+1 (o),
k
X
!
Rjk (o) Π (i, o[k|j0 ; j1 , . . . , jm ], s) , (24)
l=0
where jq = i. As we have shown above, in the event of j0 executing successfully at least
q − 1 attacks in the sequence, the expected payoff to i is not higher than the expected
payoff to i from choosing σ at o (Equation (23)). Similarly, the payoff to i in each of the
events of j0 executing successfully less than q − 1 attacks in the sequence, is not higher
than the expected payoff to i from choosing σ at o (Equation (19)). Hence in this case
Π(i, o; σ) ≥ Π(i, o, s|j0 ) as well. This shows (16).
Next we consider the case of j0 = i. Suppose that σ is a feasible sequence of i at o
maximising Π(i, o; σ) then
Π(i, o; σ) ≥ Π(i, o, s|i).
(25)
Let o0 be the state reached by successfully executing si (o) (since i ∈ Atck(s, o) so
si (o) 6= ε). If Act(o0 ) = ∅, i.e. all players choose peace at o0 , then (25) follows from
optimality of σ. Suppose that Act(o0 ) 6= ∅. Since |Act(o0 )| < |Act(o)| so, by the
induction hypothesis, for any feasible sequence σ 0 of i at o0 maximising Π(i, o0 ; σ 0 ),
Π(i, o0 ; σ 0 ) ≥ Π(i, o0 , s).
Let σ 00 = si (o), σ 0 be a sequence starting from the sequence chosen by i at
with σ 0 . Sequence σ 00 is a feasible sequence of i at o. By optimality of σ
(26)
o and ending
Π(i, o; σ) ≥ Π(i, o; σ 00 ) = pseq (i, j1 , . . . , jm )Π(i, o0 ; σ 0 )
≥ pseq (i, j1 , . . . , jm )Π(i, o0 , s)
= Π(i, o, s|i).
(27)
By (7), (16) and (27), the induction hypothesis (14) follows. This completes the inductive proof.
Step 3: For any o ∈ O(G) and any strong player i ∈ Str(G, o), an optimal full
attacking sequence is the best choice, regardless of the strategies chosen by
other players Notice that since player i is strong, she remains strong after executing
any sequence of attacks. Hence a feasible sequence σ of i at o maximising Π(i, o; σ)
must be a full sequence. This, together with Step 2 implies Step 3.
Step 4: For any player i ∈ N and any strategy profile of the other players,
s−i , such that at every state o ∈ O(G) at least one player chooses attack under
s−i , it is a best response for i to choose an optimal full attacking sequence
at every state o ∈ O(G) at which i is active, i ∈ Act(o) Take any player i ∈ N
and any strategy profile of the other players, s−i , such that at every state o ∈ O(G),
|Atck(s−i , o)| ≥ 1. Let si be a strategy of player i such that si (o) is an optimal full
attacking sequence, if i ∈ Act(o), and it is the empty sequence, otherwise. We will show
10
that at any state o ∈ O(G) such that i ∈ Act(o) and any strategy s0i of i, the expected
payoff from si to i is higher than the expected payoff from s0i ,
Π(i, o, (s−i , si )) ≥ Π(i, o, (s−i , s0i )),
(28)
with strict inequality whenever s0i (o) is not an optimal full attacking sequence.
The proof goes along the same line as the proof of a similar fact given in Step 2.
Again, we use induction on the number of active players. The induction basis is shown
by analogous arguments to those used in Step 2. For the induction step, assume that
|Act(o)| ≥ 3. By assumption, there exists an attacking player at o different to i. Let
o0 be a state reached after an attacking player is picked and a part of the whole of
her attacking sequence, τ , is executed. By the induction hypothesis, an optimal full
attacking sequence σ 0 is the best response of i at o0 . If i is not attacked under τ , then
σ 0 is the best choice at o as well. Otherwise, a permutation of τ , τ 0 , is feasible for i at
o and τ 0 , σ0 is a feasible full attacking sequence of i at o. By Corollary 3.2 it is better
than the other player executing τ first and then i executing σ 0 . It is also not better than
an optimal full attacking sequence of i. Choosing an optimal full attacking sequence is
also not worse than any other attacking sequence that i might choose. If an alternative
sequence chosen by i is full then it is obvious. If an alternative sequence chosen by i
is not full, then there is an optimal full continuation at the state it leads to and the
concatenation of the two is not worse than an optimal full attacking sequence at o.
Now take any equilibrium s. By Step 1 and Step 3, at every state there is a strong
player who chooses an optimal full attacking sequence. Hence, by Step 4, any player
chooses an optimal full attacking sequence at every state where the player is active. This
completes the proof.
3.2
Weak rewarding conflict
When γ ∈ [0, 1) then Tullock CSF is weak rewarding. As we show, for any network there
exist resource allocations such that hegemony is the only equilibrium outcome. This
requires the resources allocated to different nodes to be unequal. However, there are
networks and resource allocations such that peace is chosen by all players in equilibrium.
In this case resource allocations must be sufficiently uniform.
Possibility of peace under weak rewarding CSFs follows from two properties: CSF
is weak rewarding, so the player with lower resources has expected payoff above her
current resources and the player with higher resources has expected payoff below her
current resources. Additionally, under a Tullock CSF with γ ∈ [0, 1) it is always better
to wait for the opponents to merge, rather than to attack them in a sequence. This fact
is stated in the lemma below.
Lemma 3.4. Let p : R2++ → [0, 1] be a Tullock CSF with γ ∈ [0, 1). Then for all
x, y, z ∈ R++ ,
p(x, y)p(x + y, z) < p(x, y + z).
(29)
Since the CSF is weak rewarding, there may not exist a player who has a full attacking sequence that leads to expected resources higher than the initial resource holding.
However, if the resources are sufficiently unequal, such a player exists. More precisely,
if there exists a node endowed with resources sufficiently larger than the sum of resources in other nodes, then at every state there is a player (owning that node) who
owns sufficiently more resources than the resources of all the other players. Because of
11
that any active neighbour of that node finds it profitable to choose a full attacking sequence. Since such a player exists at every state with active players so the only possible
equilibrium outcome is hegemony.
It is interesting to note that although profitable, the full attacking sequence does not
have to be an optimal attacking sequence for the attacking player, myopically. Nevertheless it may be an optimal choice farsightedly, because the player is sure to be eventually
attacked in any equilibrium.
Proposition 3.5. Suppose γ ∈ [0, 1). There exists a resource vector r ∈ RV++ such
that for any connected network G and any state o ∈ O(G) every equilibrium outcome
contains a hegemon.
Proof. Let V be a set of nodes. Take any resource vector r such that for some node
v ∈V,
X
|V |−1
rv ≥ 2 1−γ
ru .
(30)
u∈V \{v}
Take any connected network G = (V, E) and any state o ∈ O(G). If there is a player
who owns all the nodes under o then we are done. Assume otherwise. There are at least
two active players under o, |Act(o)| = 2. Let i be the player owning node v, o(v) = i,
and let j ∈ Act(o) be any active neighbour of i under o. Let σ be a permutation of
Act(o) \ {j} starting with i. Sequence σ is a full attacking sequence of j at o. We will
show that Π(j, o; σ) > Rj (o). Let x = Rj (o) be the resources of j at o and y = Ri (o)
be the resources of i at o. By our assumptions,
y≥2
|V |−1
1−γ
x.
(31)
After winning a conflict with i, in every subsequent conflict in the sequence j has higher
resources than her opponent. Hence the probability of winning each these conflicts is
more than 1/2. In the event of winning all the conflicts in the sequence, player j owns
at least x + y resources. By these observations
1
xγ
Π(j, o; σ) ≥
(x + y),
(32)
2m−1
xγ + y γ
where m = |Act(o)| is the number of active players at
By (31), q + 2 − m ≥ γq + 1, hence
2q+2−m ≥ 2γq+1 > 2γq + 1 −
This can be rewritten as
o. Let q = (|V | − 1)/(1 − γ).
1
2m−2
1 + 2q
≥ 2m−2 .
1 + 2γq
.
(33)
(34)
Since h(z) = (1+z)/(1+z γ ) is increasing on R++ for γ ∈ [0, 1) so, by (31), the inequality
above implies
1 + xy
m−2
(35)
y γ ≥ 2
1+ x
which can be rewritten as
1
2m−1
xγ
xγ + y γ
12
(x + y) > x.
(36)
Thus
Π(j, o; σ) > x
(37)
and so choosing σ player j strictly increases her expected payoff.
Since at every state o with active players there exists a player who can increase her
expected resources by choosing attack, so every equilibrium outcome is hegemony.
Peace in the first round is possible for resource allocations which are sufficiently close
to the uniform allocation. We illustrate that with two examples: one on a clique over
four nodes and one on a star over for notes. It is worth noting that resource allocations
given in the examples require sufficiently low values of γ to allow for peace under the
strategy profile provided. Moreover, the restrictions on γ are less stringent in the case
of the sparser network.
Example 3.6 (Peace on a clique). Let V = {1, 2, 3, 4} and let G be a clique over V and
let N = {p1 , p2 , p3 , p4 } be the set of players. Suppose that initially, at state o, every
player owns exactly one node, o(i) = pi . Consider the following strategy profile. At state
o0 every player chooses the empty sequence ε. At any state with three active players
where nodes i and j are owned by player a, player a attacks her neighour b and player
b attacks a; the remaining player chooses peace. At any state with two active players, a
player with minimal resources attacks the other player. The strategy profile is illustrated
in Figure 2. Under this strategy profile, any player deviating from peace in the initial
state, will either loose a conflict or will be engaged in three conflicts. Moreover, the
construction ensures that the strategy profile in states with at most three active players
is an equilibrium. Therefore, if the resource endowment is such that engaging in three
conflicts is not profitable for any player, then s is an equilibrium.
Figure 2: Schematic illustration of strategy profile described in the example. Red arrows
indicate bilateral conflict. Nodes owned by the same player are encircled with thicker
lines.
If the resource endowment is r ∈ {(1, 1, 1, 1), (1, 1, 1, 2)} then s is en equilibrium
for all γ ∈ [0, 1). When r = (1, 1, 1, 3) then s is an equilibrium for all γ ∈ [0, 0.45)
(and for γ ∈ [0.46, 1) it is not an equilibrium). When r = (1, 1, 1, 4) then s is an
equilibrium for all γ ∈ [0, 0.15) (and for γ ∈ [0.16, 1) it is not an equilibrium). When
r ∈ {(2, 1, 1, 3), (3, 1, 1, 2)} then s is not an equilibrium for all γ ∈ [0, 1).
Example 3.7 (Peace on a star). Let V = {1, 2, 3, 4} and let G be a star network
over V with centre 1 and let N = {p1 , p2 , p3 , p4 } be the set of players. Suppose
13
that initially, at state o, every player owns exactly one node, o(i) = pi . Consider
strategy profile s constructed in Example 3.6. If the resource endowment is r ∈
{(1, 1, 1, 1), (1, 1, 1, 2), (1, 1, 1, 3), (1, 1, 1, 4)} then the thresholds for γ under which s is
an equilibrium are as in Example 3.6. When r = (2, 1, 1, 2) then s is an equilibrium for
all γ ∈ [0, 0.87) (and for γ ∈ [0.88, 1) it is not an equilibrium). When r = (2, 1, 1, 3) then
s is an equilibrium for all γ ∈ [0, 0.33) (and for γ ∈ [0.34, 1) it is not an equilibrium).
When r = (3, 1, 1, 2) then s is an equilibrium for all γ ∈ [0, 0.23) (and for γ ∈ [0.24, 1)
it is not an equilibrium).
4
Conclusions
We studied a model of dynamic conflict on a network. We showed that if the Tullock
contest success function determining the winner of a bilateral conflict increases expected
resources of the stronger player (Tullock parameter γ > 1) then in every equilibrium
outcome there is one player owning all the nodes in the network. Moreover, at every
state there is a player who finds it profitable to fight, regardless of the choices of others. In the case of Tullock contest success function increasing expected resources of the
weaker player (Tullock parameter γ ∈ [0, 1)), we showed that if resource endowment is
sufficiently unequal, then in every equilibrium outcome there is one player owning all resources. With sufficiently uniform resource endowment, peace is possible in equilibrium,
as illustrated by examples.
Acknowledgements. Marcin Dziubiński acknowledges support from Polish National
Science Centre through grant nr 2014/13/B/ST6/01807. Sanjeev Goyal acknowledges
support from a Senior Keynes Fellowship. Sanjeev Goyal and David Minarsch acknowledge support from the Cambridge-INET Institute.
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15
Appendix
Proof of Lemma 3.1. Since γ > 1 so the function h(x) = xγ is strictly convex and
h(0) = 0. By strict convexity of h, for any y, z ∈ R++ , h(y) − h(0) < h(x + y) − h(x)
and, since h(0) = 0, so h(x + y) > h(x) + h(y). Thus for any y, z ∈ R++ ,
(y + z)γ > y γ + z γ .
(38)
Take any x, y, z ∈ R++ . Multiplying both sides of (38) by (x + y)γ we get
(x + y)γ (y + z)γ > (x + y)γ (y γ + z γ ).
(39)
Since, by (38), (x + y)γ > xγ + y γ so
(x + y)γ (y + z)γ > (x + y)γ y γ + (xγ + y γ )z γ .
(40)
Adding xγ (x + y)γ to both sides we get
(x + y)γ (xγ + (y + z)γ ) > (xγ + y γ ) ((x + y)γ + z γ )
which can be rewritten as
(x + y)γ
1
1
> γ
.
xγ + y γ
(x + y)γ + z γ
x + (y + z)γ
Multiplying both sides by xγ we get
xγ
(x + y)γ
xγ
>
.
xγ + y γ
(x + y)γ + z γ
xγ + (y + z)γ
(41)
(42)
(43)
This completes the proof.
Proof of Corollary 3.2. We start by showing, for any m ≥ 3 and x1 , . . . , xm ∈ R++ that
!
m
X
pseq (x1 , x2 , . . . , xm ) > pseq x1 ,
xl .
(44)
l=2
Proof is by induction on m. The induction basis, m = 3, holds by Lemma 3.1. For
induction step, suppose that m > 3 and that the claim holds for all 3 ≤ m0 < m. By
the induction hypothesis and Lemma 3.1,
pseq (x1 , x2 , . . . , xm ) = p(x1 , x2 )pseq (x1 + x2 , x3 , . . . , xm )
!
m
X
> p(x1 , x2 )pseq x1 + x2 ,
xl
l=3
= pseq
x1 , x2 ,
m
X
!
xl
l=2
> pseq
x1 ,
m
X
l=2
This completes the inductive proof.
16
!
xl
.
(45)
For the main claim of the lemma, by what was shown above,
pseq (x1 , . . . , xi−1 , xi , . . . , xj , xj+1 , . . . , xm ) =
!
i−1
X
pseq (x1 , . . . , xi−1 )pseq
xl , xi , . . . , xj pseq
pseq (x1 , . . . , xi−1 )pseq
l=1
!
xl , xj+1 , . . . , xm
>
l=1
l=1
i−1
X
j
X
xl ,
j
X
!
xl
pseq
j
X
!
xl , xj+1 , . . . , xm
=
l=1
l=i
pseq (x1 , . . . , xi−1 , xi + . . . + xj , xj+1 , . . . , xm ),
(46)
(in the case of empty sequence we assume pseq (ε) = 1). This shows that Equation (10)
holds.
Proof of Lemma 3.4. Since γ < 1 so the function h(x) = xγ is strictly concave and
h(0) = 0. By strict concavity of h, for any y, z ∈ R++ , h(y) − h(0) > h(x + y) − h(x)
and, since h(0) = 0, so h(x + y) < h(x) + h(y). Thus for any y, z ∈ R++ ,
(y + z)γ < y γ + z γ .
(47)
The remaining part of the proof is analogous the that in proof of Lemma 3.1, with
all inequalities reversed.
17