A Dynamic Model of Conflict and Cooperation

stock Z in a given period. To model this probability for agent i, a natural assumption is that
the probability is increasing in aggressive investment of agent i, the fraction of time player i

devotes to aggression, and decreasing in the sum of aggressive investment of all agents. A
plausible form of the conflict technology is the Tullock contest success function (Tullock,
1980; Hirshleifer, 1991 and 1995; Gonzalez, 2007). In its standard formulation this function

reads:

where the parameter r captures the effectiveness of aggression. From the contest success
function (4) we obtain the relative success of contender i in the contest. Alternatively, the
contest success function (4) may be interpreted as a sharing rule, or ownership of assets that
depends on the respective efforts of aggression. It is natural to assume in the analysis that
each agent has an equal access to the prize when agents do not engage in aggressive behavior;
hence the assumption that p_i (0,...,0) = 1/n will be in force throughout the analysis.

The instantaneous expected payoff to each agent is given by pi (a1,...,an) Z.⁵ Each of the
agents chooses the streams of a_i and l_i to maximize the discounted value of total expected
payoffs subject to the feasibility conditions introduced in (1)-(4):

Z = XX (1 - aj) - δZ, Z (0) = Zo ≥ 0,

j=1

0 ≤ ai (t) ≤ 1 for   ∀t ∈ [0, ∞) ,

where ρ > 0 is the rate of time preference.

⁵ Alternatively, one may view the prize as flow services, such as output or utility from the stock variable Z
rather than Z itself. To clarify this we need to introduce a concave function, say u (Z) instead of Z. However,
this complication does not affect our results at all.

<<   6   7   8   9   10   11   12   >>

More intriguing information