if the resource enters into a conflict situation, then there exists a positive probability that at the
end of the period it will be the property of someone else.
We define the set of indexes I = {i ∈ N* :1 ≤ i ≤ I} and identify each agent by her corresponding
index i ∈ I. We assume that the time is discrete, and the conflict has a time duration of T ∈ N*
periods. Let T = {t ∈ N* :1 ≤ t ≤ T}be the set of discrete temporal indexes.
We model each conflict event as a one-time-played game with dependence on past events. That
is, each period a new game is played, so each time there will be a new assignation of the divisible
resource, but the state of the conflict will depend on past choices. We have that the number of
games played T → ∞ . Then we do not rule out, by assumption, all non-competitive strategies3,
but we focus in non-cooperative games.
Each agent i ∈ I has an initial valuation scale for the good that she is able to obtain. We represent
an I-dimensional valuation vector vI = (vQ,..., vI) > 04. Agents’ initial value scale are drawn from
Ψ = {vQ∈ [v, v] : v ≥ 0, (∀i ∈ I)} .
We assume that each agent has the incentive to exert some effort to the conflict in order to obtain
a proportion of the divisible resource: each agent receives utility from her subjective valuation of
the good and from the quantity of the valuable resource that is left to use freely. Additionally we
allow agents to invest in conflict technology.
The quantity cit is the amount of resource available for free use and is the quantity that the agent
i ∈ I has in the period t ∈ T, after all other uses have realized (that is, after the cost incurred to
obtain the valuable resource and the investment are realized). That is determined by
ct = μtRt - gi (et; θt) - xt
where μt is the share of the good obtained, Rt is the total amount of the resource available in the
period t ∈ T, ett is the effort used explicitly in the conflict by agent i, gi (ei; θ'j is the cost (in units
3The finiteness of the repeated game would allow us to solve the game by backward induction. This would allow
us to rule out, by assumption, all non-competitive strategies. For if such strategy exists, some player has the
incentive to deviate from it in order to obtain a higher discounted utility. If so, that player would deviate in
the last period, to avoid punishment. But knowing that, all other agents would also deviate from the strategy.
Solving for the period T — 1 we would find the same situation. Then, the finiteness of the game, that impedes a
credible punishment to the players who deviate from a possible cooperative strategy, would allow us to rule out
this possibility.
4As standard in the economic literature (in the convex analysis literature) we define the following order relation for
vectors.
Let u, v ∈Rn be two vectors, and consider the convex cone R ". = {r = (r1,...,rn) ∈ R ". : r1 > 0,... ,rn > 0}.
Then we define the order relation > as:
u > v → u — v ∈R++