Conflict and Uncertainty: A Dynamic Approach

process affected by random shocks

Zt = z (zt-ι,ez)

We also assume that the cost parameters θ_t = (θ_t¹, θ2,... θI) are random variables.

Let

F (Rt,e^z_t ,θt)

be the continuous cumulative joint distribution function of resources and technological characteristics
of the agents. All the marginal distributions derived from F (∙) are also continuous. We assume that
each agent i ∈ I knows her own cost parameter vector θⁱ that is drawn according to F (∙) during
each event, before the choice-making process takes place. But the agent is unaware of the vectors
θI∖{i} = (θ_t¹,..., θt^-1,θt⁺¹,..., θI), the cost vectors associated to all the other agents of the conflict.
This reflects the fact that the relative strength of the opponents is unknown during each event, and
in this context we treat it as a random variable. Another approach could try to endogenize the
evolution of the effort-cost factors, with investment in conflict machinery, investment in research and
development or adaptation to the conflict situations (as proposed by Weeks (2001)). Nevertheless
this could neglect factors such as luck, climate and geographic influences or an exogenous change
taking place in the conflict. Then, as a first approach we prefer to model the effort-cost coefficients
as an exogenous process and focus on the role of uncertainty on the conflict costs. And later we will
focus in investment in conflict technology, not in effort costs.

We assume rational expectations: the agents take all available information into account in forming
expectations. Formally we assume that the agents know the associated distributions for the random
variables. This is equivalent to assume that the subjective distributions taken to form expectations
are the same objective distributions from the conflict environment.

3.4 This is an incomplete information game

For clarification we summarize the information using a version of game theory axiomatization. Our
incomplete information game (ω,φ,σ) has the following components:

1. A set of private information ω = {α,θ,π, g} where αⁱ = {αt}_t∈τ ^aɪɪd α = {αⁱ}i∈I, α is a
TI × 1 vec tor, θ = {θⁱ}i∈I is a I × 1 vec tor, g = {gⁱ}_i∈I a nd π = {πⁱ}_i∈I a re I × 1 vectors.

2. A set of possible messages φ = {^ⁱ}_i∈I a nd φ^l : ω → R+ that maps from the private
information set to the real set, that is φ = {e, x} is a 2 (TI × 1) vector.

3. A set of common information σ = {σ_t}_t∈_τ = {μ, I, T, R_t-1,z_t-1}. That is, an a !location rule μ
that says how the game works, I agents, T periods, the available resource under dispute in t — 1 and

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