More of The Game of "Chicken" and Cold War Nuclear . . . by Steve Lee
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Using mixed-strategies Nash Equilibria as an alternative method(continued from previous webpage)
If we recall from an earlier discussion, the mixed-strategies Nash Equilibrium for this "Chicken" game is (1/2, 1/2) = (Cooperate, Not Cooperate) for both players. The mixed-strategies equilibrium, unlike the pure-strategies equilibrium, invloves alternat ing strategies over time. Therefore, '1/2' denotes the probability for each player of making a particular move. For example, both player 1 and player 2 will play -- in the long run -- half the time 'Cooperate' and half the time 'Not Cooperate' (but they d on't necessarily play the same strategy at the same time -- although they could do so).
If we find the expected payoffs for playing an alternating pure-strategies equilibria -- (Cooperate, Not Cooperate) and (Not Cooperate, Cooperate) -- exclusively, then we will find that for both player 1 and 2, that their expected value will be a p ayoff of 2 ( E(Payoff for playing pure-strategies)=(Probability of Cooperation)*(Payoff for Cooperation given other player plays 'not cooperate')+(Pr. of Not Cooperating)*(Payoff for Not Cooperating given other player 'cooperates') -- theref ore, for player 1 and 2, E(Payoff)=(1/2)*(1)+(1/2)*(3)=2).
If the expected value of payoffs for playing alternating pure-strategies Nash Equilibria is 2, then the expected payoff is the same as the (Cooperate, Cooperate) solution (this result should be the same for other sets of payoffs that follows the "C hicken" game format). In other words, alternating pure-strategies over time gives you no better or no worse a result than if both sides cooperated (although with differing sets of payoffs, this statement may not hold completely). We can be sure that -- at worse -- the expected values of alternating pure-strategies as well as the payoffs from the cooperative solution should be higher than the payoffs for the non-cooperative solution (with other sets of payoffs, we can be at the very least sure that the preceding statement would be true).
These results suggest a possible solution to the dilemmas posed by a game theoretical model of nuclear conflict: rational parties will cooperate when faced with a scenario such as this. Why would that be the case? Since the expected payoffs for alternatin g pure-strategies over time is the same as the payoffs for a sustained cooperative solution, both sides may find it more efficient to skip the 'tit-for-tat' and nuclear posturing and go directly to mutual cooperation. Another adavantage to playing the fully cooperative solution is that neither side will run the danger of playing 'Not Cooperate' when the other side plays 'Not Cooperate' (which is possible by playing mixed-strategies over time). The results of that possibility would be catastrophic f or both sides.
To reiterate, by both sides playing the game of 'nuclear chicken' at where the expected values of alternating pure-strategies over time is equal to the payoffs of a possible solution (or choosing a solution that has the payoffs closest to the expec ted payoffs -- especially if a mutually catastrophic solution is possible without resorting to this stratagem (this will only apply if we have sets of payoffs that do not conform to the relationships found here)), we can have a rationally op timal way of achieving cooperation and preventing a nuclear disaster.
Empirically speaking -- at least with the Soviet Union versus the United States -- we have seen no exchanges of nuclear weapons between the two sides. There has been posturing and threats, but these events have never turned into an all out nuclear confron tation (it should be noted that the idea of adding 'threats' to a model is useful, but, ultimately, unsatisfactory to determining what will be the major outcome of confrontations (e.g., will the two sides fight because of threats or merely bluff away from a fight?)).
Rational players -- who have done the necessary calculations in at least a sub-conscious way -- will probably back away from a 'chicken' scenario that emperils so much of our world. However, the author cannot say for certain that an irrational player will "Cooperate" in the attempt to avoid all out nuclear destruction.
The End.