Nash's Equilibrium: Unequal Starting Conditions & Understanding

[dratsab]

So, if both sides of a war have chosen the best war strategy in advance, however, one side has less troops, and therefore loses... is it still considered a state of Nash Equilibrium even though the starting conditions weren't equal? Also, if anyone is willing, I like the idea of Nash Equilibrium, but I don't have the mathematical intelligence to understand the equation for it, if anyone is willing to dumb it down to where I understand what it all means, that would be appreciated.

 

[JSuarez]

Suppose that you are in a game with just one other player (the extension to several players is starightforward; the restriction is for clarity only) and have a set of strategies, from among you can choose one; the same is true for the other player.

Now, suppose that the other player chooses a strategy from his set; among your strategies there will be a subset that is the best response to the other player. This is not yet a Nash equilibrium, because the other player may change his strategy, to another one to which yours is no longer a best response but, if the other player's strategy is a best response to yours (this means that both strategies are best responses to each other), then you are at a Nash Equilibrium, in the sense that neither of you has any incentive to change strategies.

Consider the famous example of the Prisioner's Dilemma: if they both stay silent, they get the high reward; if they both confess, they both get a much lower reward and, if just one confesses, he gets the hightest reward, while the other gets nothing.

Where is the equilibrium? It's not the "stay silent" strategy, because both have an incentive to change. It's not either of the pairs (confess, stay silent), because the "stay silent" one has an incentive to change. It leaves the "confess": in this case, neither one has any incentive to change,

 

[dratsab]

Suppose that you are in a game with just one other player (the extension to several players is starightforward; the restriction is for clarity only) and have a set of strategies, from among you can choose one; the same is true for the other player.

Now, suppose that the other player chooses a strategy from his set; among your strategies there will be a subset that is the best response to the other player. This is not yet a Nash equilibrium, because the other player may change his strategy, to another one to which yours is no longer a best response but, if the other player's strategy is a best response to yours (this means that both strategies are best responses to each other), then you are at a Nash Equilibrium, in the sense that neither of you has any incentive to change strategies.

Consider the famous example of the Prisioner's Dilemma: if they both stay silent, they get the high reward; if they both confess, they both get a much lower reward and, if just one confesses, he gets the hightest reward, while the other gets nothing.

Where is the equilibrium? It's not the "stay silent" strategy, because both have an incentive to change. It's not either of the pairs (confess, stay silent), because the "stay silent" one has an incentive to change. It leaves the "confess": in this case, neither one has any incentive to change,

But Nash would argue that you shouldn't do this, correct? The idea is to realize you can achieve more by working together, that greed is not good, kind of like the 2nd zeitgeist video talks about. I get that I think, but what if there is no incentive to change even if losing? What if the best strategy you can pick results in loss?

 

[CRGreathouse]

But Nash would argue that you shouldn't do this, correct? The idea is to realize you can achieve more by working together, that greed is not good, kind of like the 2nd zeitgeist video talks about. I get that I think, but what if there is no incentive to change even if losing?

Regardless of what you 'can' or 'should' do, both players defecting is the Nash equilibrium.

What if the best strategy you can pick results in loss?

If every combination of choices for every player results in the same outcome for you, then it doesn't matter what you do -- this isn't game theory, at least for that player. But if your best strategy results in loss rather than, say, ruination then it's just a normal game.

 

[CRGreathouse]

So, if both sides of a war have chosen the best war strategy in advance, however, one side has less troops, and therefore loses... is it still considered a state of Nash Equilibrium even though the starting conditions weren't equal?

It could be -- Nash equilibria aren't about fairness or equality of starting conditions. What makes it a Nash equilibrium is that neither player wishes to deviate, given the other's strategy.

 

[JSuarez]

But Nash would argue that you shouldn't do this, correct?

No, you are conflating two different things: the Nash equilibrium and the Nash bargaining solution; the former, as CRGreathouse pointed out, has nothing to do with fairness or cooperation (in fact, it's usually defined in the context of non-cooperative games); the latter is intended to be a model of "fairness", or distributive justice, but it can be defined (by the Nash-Zeuthen bargaining axioms) independently of the Nash equilibrium.

The idea is to realize you can achieve more by working together, that greed is not good, kind of like the 2nd zeitgeist video talks about.

This is a moral philosophy question. This equilibrium concept doesn't have anything to say about these.

What if the best strategy you can pick results in loss?

In fact, this happens in a lot of game models, both theoretical, where you may have several distinct equilibriums and no way of telling which one will be picked, and experimental, where it has been found that most people don't behave rationally, or is even capable of finding the best solution. An extreme example is the zero-sum games, where a player's gain is another player's loss; in this case, the Nash equilibrium always entails a loss for someone (or zero gain for both).

 

[SW VandeCarr]

So, if both sides of a war have chosen the best war strategy in advance, however, one side has less troops, and therefore loses... is it still considered a state of Nash Equilibrium even though the starting conditions weren't equal?

Your conclusion, "...and therefore loses" doesn't necessarily follow. There are certainly successful best strategies given inferior numbers. Look up Lee's defense of Richmond in the 1862 Peninsula Campaign of the American Civil War (or the 1863 Battle of Chancellorsville).