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.