RPGnet Columns
11-03-2003, 10:19 PM
Post originally by Steve at 2003-11-03 21:19:22
Converted from Phorums BB System
I can't believe you have an essay of this length, and cover optimization routines, but you leave out the concept of Nash equilibria. Once you bring this concept in even games where the strategy space is a subset of a continuous space (such as the real numbers) you will find an equilibirum. For one shot games, finding an equilibrium is not that tough. Some games might have multiple equilibria, but the number is still usually finite.
[A Nash strategy is where a player plays a strategy such that given the other players strategy (whatever it is) you take the action that gives the best payoff. Since game theory assumes both (all) players are rational (how do you model irrational behavior mathematicall!?!) the other player will also select a Nash strategy. Selecting Nash startegies results in a Nash equilibrium. This simple, yet brilliant idea was what lead to Nash sharing the Nobel prize in economics. The concept was in Nash's PhD dissertation and IIRC, was very short, something like 42 pages.]
The real problem is with games that are infinitely repeated (or where the last round of play is not known with certainty). In this case, the folk theorem enters in often times resulting in an enormous increase in the number of equilibria. Still there are some refinements that can help narrow down the number of equilibria. For example, there is the concept of sub-game perfection. This where if a strategy is an equilibrium in each sub-game of the repeated game, then you might want to consider that equilibrium as being some what "better" than non-sub-game perfect equilibria.
Then there are focal points which is where intuition is used to figure out if some equilibria are more likely than others. And finally there is the research that looks at learning and experimental evidence. For example, most people generally use simple decisions rules vs. something like Bayes theorem.
So the idea that because the number of strategies is large you cannot find a solution is I think a bit misleading. Looking for Nash equilibria, especially in a one shot game will solve alot of the problem.
Converted from Phorums BB System
I can't believe you have an essay of this length, and cover optimization routines, but you leave out the concept of Nash equilibria. Once you bring this concept in even games where the strategy space is a subset of a continuous space (such as the real numbers) you will find an equilibirum. For one shot games, finding an equilibrium is not that tough. Some games might have multiple equilibria, but the number is still usually finite.
[A Nash strategy is where a player plays a strategy such that given the other players strategy (whatever it is) you take the action that gives the best payoff. Since game theory assumes both (all) players are rational (how do you model irrational behavior mathematicall!?!) the other player will also select a Nash strategy. Selecting Nash startegies results in a Nash equilibrium. This simple, yet brilliant idea was what lead to Nash sharing the Nobel prize in economics. The concept was in Nash's PhD dissertation and IIRC, was very short, something like 42 pages.]
The real problem is with games that are infinitely repeated (or where the last round of play is not known with certainty). In this case, the folk theorem enters in often times resulting in an enormous increase in the number of equilibria. Still there are some refinements that can help narrow down the number of equilibria. For example, there is the concept of sub-game perfection. This where if a strategy is an equilibrium in each sub-game of the repeated game, then you might want to consider that equilibrium as being some what "better" than non-sub-game perfect equilibria.
Then there are focal points which is where intuition is used to figure out if some equilibria are more likely than others. And finally there is the research that looks at learning and experimental evidence. For example, most people generally use simple decisions rules vs. something like Bayes theorem.
So the idea that because the number of strategies is large you cannot find a solution is I think a bit misleading. Looking for Nash equilibria, especially in a one shot game will solve alot of the problem.