I watched a short about John Nash, the Nobel Prize winner for his work on Game Theory. The short explained briefly that it is the concept of anticipating what your opponent will do influences what you do, and I understand how that can be applied to economics, but only in the most basic way. They also said that Nash proved this with mathematics. I don’t understand how that’s even possible (not that I doubt it, mind you). Can someone explain how that was done? I need the most basic explanation, as my understanding of advanced mathematics only stretches to the concept of balancing one’s checkbook (and I’m not getting high marks in that, either). Thank you for typing v-e-r-y s-l-o-w-l-y.
Imagine a game with two players, each choosing from two options, A and B. Set up a 2×2 matrix, with player 1 on the, let’s say, vertical side, and player 2 on the horizontal. Each has two options, A1 and B1, and A2 and B2, respectively. Now, go to the (A1,A2) cell; that’s the outcome to each player if they each choose those options. Inside the cell might be, let’s say, money payoffs: $0 for 1 and $1 for two.
Now, player 2 gets $1 if player 1 chooses A1 and player 2 chooses A2. Go to the (A1, B2) cell. Suppose the payoff there is $1 for player 1 and $2 for player 2. Then, player 2 gets $2 if player 1 chooses A1 and player 2 chooses B2. So, if player 1 chooses A1, player 2 will want to choose A2. Underline player 2’s payoff of $2, since that’s what he wants when player 1 chooses A1.
Repeat player 2’s options with player 1 choosing B1. Underline player 2’s preferred payoff.
Repeat with player 1’s options as player 2 chooses A2 and then B2. Underline for player 1.
In the cells where both players’ payoffs are underlined are Nash Equilibriums. A Nash Equilibrium is where no player wants to change his strategy based on the payoffs. If no cell has both payoffs underlined, then there is no Nash Eqm, and if multiple cells have underlines, then there are multiple Nash Eqms.
If you look at the driving game here, you can see that there is clearly two Nash eqms: everybody drives their cars on the right or the left. History decides which particular equilibrium obtains.
Does that help?
More generally:
You have A actors and each actor has Ca potential actions. There is a function which takes as it’s inputs all the actions by the actors and turns out a state of the world form a set ofa ll available states. Each actor has a utility type function which takes the states of the world and gives a utility value (actually you can start back with that the actors simply need a set of ordinal preferences over the states of the world).
These are the given parameters, the tricky part is finding whcih conditions on these assumptions would bring about an equilibrium (things like continuity of the function, covexity of the preferences, assumpitons about spaces, etc.), generally a point where every player is making a decision from which they cannot improve themselves by picking an alternative action. In the Nash case you then can show that there is a state of the world that leaves everybody better off then the equilibrium case.
It actually can involve some pretty complex calculus, topography, set theory etc.
Okay, I re-read the OP and I think I may have misunderstood the question.
How, mathematically, would one prove that an equilibrium exists? Using a fixed-point theorem. Very, very roughly, a fixed-point theorem establishes that in a topological space with certain characteristics, if you move every point to another spot, there will be a point that stays in the spot it is in. It’s a mathematical way of saying “We know there’s a needle in that haystack,” without necessarily knowing where it is.
So, for n players, player1 must come up with a best strategy for each possible grouping of the n-1 other strategies. It turns out that there is a n-1 strategy set that provides player1 a best response to them. You’re taking the set of n-1 strategy sets and mapping it onto the set of best responses based on the n-1 strategy sets, and the fixed-point theorem tells you that at least one n-1 strategy set is the best response for it. Since it is general, it applies to all players.
That’s the best I can do. Sorry!
Um…thanks. I kind of get it. I appreciate your efforts, but this stuff is just waaay hard for me. I’ll have to ponder it some more.
A fixed point theorem has one other important characteristic, namely that the deformation must be continuous. In layman’s terms, a continuous deformation means that we’re not doing anything drastic such as ripping the space apart.
The most famous one, the Brouer fixed-point theorem, covers spheres. Suppose I start with a ball of silly putty. I mash it, squish it, squeeze it, stretch it, bend it, and so forth, and then form it back into a sphere. There’s some molecule of silly putty that’s in the same place that it started in. On the other hand, if I do something drastic like cutting the lump into a thousand little pieces and sticking those pieces back together, there may not be a fixed point.
The Wikipedia entry on Nash Equilibria is not bad.
Game theory is tricky, not least of all because it assumes that the actors are rational beings which means that its applicability to “real” life is strictly limited. A lot of the math is just enumerating the possibilities for each actor (and then generating proofs describing their optimal actions).
One thing that’s important to note is that the other person is not necessarily your opponent. There are lots of types of “games”, not all of which have winners and losers. In the Nash equilibrium that other posters have mentioned, the goal of the game is to get each “player” to the point where if they * unilaterally * make a different decision, their payoff does not increase. Now note that this says nothing about other players payoffs, nor does it say what happens if two players cooperate in changing their strategies. However identifying the Nash Equilibrium points in a game can be useful if you can make assumptions about your opponents. (See the “Occurrence” section in the Wiki article.)
This illustrates that much of game theory depends on the initial assumptions of the game (which are often very rigid or unrealistic in the real world).
So consider some of the assumptions:
a. Does each player know their opponent’s pay-off function?
b. Are players allowed to cooperate?
c. Are players antagonistic (actively trying to minimize the other guy’s payoff), altruistic (trying to maximize the overall game payoff), or self-interested (trying to maximize their own payoff)?
d. Is the game a “one-shot” or is there a continuing relationship between the actors? In games such as the Prisoner’s Dilemma, the best results for all parties can occur when the parties cooperate and choose payoffs that are compromises. However if one party defaults (and screws the other, e.g. ratting on a compatriot), they can get a better payoff (e.g. getting a pardon) while eliminating any chance of trust in a future game.
e. Are all players rational and intelligent enough to compute the payoff functions? (Look at some EBay auctions and you’ll realize that this would be an erroneous assumption in real life.)
It may be helpful for you to know that Nash contributed to game theory, but that the concept of game theory itslef had already around for a little while. In other words, there is a difference b/t describing “Nash Equilibium” and “Game Theory”.
You may already be aware of this, but I thought I’d try to help anyway.