Conflict and Uncertainty: A Dynamic Approach

The parameter α^l_t represents a measure of ex-ante relative effectiveness in the conflict: ceteris
paribus, higher levels of α^l_t increase the share of the good obtained by agent i. This technological
parameter can be influenced by investment in conflict technology. In particular, we have a transition
function

^αt+ι    ^α (^xt, ^αt, ^zt, ^st)

satisfying ∂χ∙αⁱ (xt,⅛z_t) ≥ 0 (rising investment level x^l_t will never decrease the technological
parameter function αⁱ), where st iɛ ^a success measure for the agent i ∈ I in the period t ∈T. The
factor z_t is an exogenous shock to investment in conflict technology.

This investment in conflict technology is a key ingredient in dynamic conflicts. For example, since
the “enemy adapts to the methods employed by the attacker” (Weeks (2001)) we could think of
a change in the relative effectiveness of the agents according to the average success on each time
period. In that case, the weaker players could adapt to the harder environment, and increase their
relative effectiveness. This would imply a negative relationship between success and next-period
effectiveness. This fact could explain why ex-ante weak fighters can resist fights over long periods.
This cannot be modeled without an explicit dynamic mechanism.

We represent the preferences on the net share of the resource obtained with a utility function

πi = ∏ ,u (ct)) : R+ → R

that is increasing (at a decreasing rate) in its arguments, where vt is the subjective valuation for the
good in dispute (is the value scale parameter when πⁱ is multiplicative!y separable) and u^l (c^l_t) is the
standard instantaneous utility function. A higher valuation for the good in dispute implies that the
utility obtained by consumption of the good in dispute is more valuable to the agent. We assume
that a higher quantity of the good left for free use increases the utility received by the agent with
decreasing marginal returns. We impose the standard conditions on the utility function: ∀i ∈ I,
uⁱ (∙) a nd πⁱ (∙) are continuously differentiable concave functions.

3 Simple Characterization

3.1 Allocation Rule

Given the effort level e¹ and the effectiveness parameters αI, the allocation rule (the central planner,
the government, nature or the justice or whatever in what you believe) assigns the resource in dispute

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