Page:CIAdeceptionMaximsFactFolklore 1980.pdf/38

C00036554 FIGURE 5: A CONCISE STATEMENT OF AXELROD'S GAME

1. The player is presented with an infinite sequence of opportunities, i=1,2,...

2. When an opportunity is presented the player can elect to use the resource, and receive a value, $$E \cdot x_i$$, where $$E$$ is a known constant (the enhancement factor) and $$x_i$$ is the outcome or value of the i-th opportunity. Alternatively, the player can wait and defer a decision until the next opportunity, in which case a cost, $$-x_i$$, must be paid.

3. If the the resource is "used" on any opportunity, there is a probability, Q, that it "survives" and can be used again, and 1-Q that the game will terminate.

4. If the resource is "saved" on any trial, there is a probability, D, that the game will terminate, and 1-D that it will continue until the next trial. Equivalently, D can be viewed as a discount factor from trial to trial.

5. The values on successive trials are independent with known and common density function, f(x).

6. The optimal policy is to define a threshold, t, and use the resource if $$x_i \geq t$$, otherwise to save it.

7. The value of the game, V(t*), and the optima1 threshold, t*, can be determined by univariate optimization of the function:

$$ V(t^*) = \max_t V(t) = \frac{E\cdot p(t) \bar{S}(t) - (1-p(t)) \underline{S}(t)}{D+(1-D)(1-Q)p(t)} $$

where

$$p(t) = \int_{t}^{\infty} f(x)dx$$,

$$ \bar{S} = \int_{t}^{\infty}xf(x)dx$$

and

$$\underline{S} = \int_{0}^{t}xf(x)dx$$

for continuous distribution or appropriate sums for discrete distributions.