It is clear that players of a game can almost always create trivial variations on strategies without breaking game theoretic conclusions. For example, a player playing Rock Paper Scissors can play any of...
- Throw rock and blink
- Throw rock and do not blink
- Throw paper and blink
- Throw paper and do not blink
- Throw scissors and blink
- Throw scissors and do not blink
Game theory would be pretty useless in the real world if solutions to the "new" game this creates could not be analyzed by partitioning this strategy space into the superstrategies...
- Rock strategies
- Paper strategies
- Scissors strategies
...and applying the tools used to solve the ordinary version of Rock Paper Scissors to these superstrategies (at least up to some imperceptible negative utility of blinking). Can we, in general, less trivial contexts, port the tools of game theory ordinarily applied to strategies (dominant and dominated strategies, iterative deletion, best responses, Nash Equilibria and other solution concepts, backward induction, etc.) over to these superstrategy partitions of a strategy space?
Example #1: In a trading card game, a player builds a custom deck of cards to battle another player with. Suppose there are $5000$ cards available, and each deck must contain $50$ unique cards. Then the player has $5000 \choose 50$ possible pure strategies for building a deck or mixing over. Further suppose we come up with an algorithm which categorizes each strategy as either aggressive (meaning the probability of the resulting deck [or expected deck if mixing] winning a battle is higher in shorter games; it is suited to the role of expending resources ambitiously to seek a fast victory), or passive (meaning the probability of the resulting deck or expected deck winning a battle is higher in longer games; it is suited to the role of waiting for the opponent to expend resources before seeking to win). We could apply any of the usual tools of game theory to strategies, but the strategy space is huge. The approach under consideration in this question is to apply the same tools to the superstrategies...
- Build an aggressive deck
- Build a passive deck
For example, a superstrategy profile might be considered a Nash Equilibrium iff both players with common knowledge of whether the other has chosen an aggressive deck or a passive deck still do not benefit by switching their own deck from aggressive to passive, or vice-versa. They are allowed to benefit by switching to a different strategy within the same superstrategy (i.e., switch from one aggressive deck to a different aggressive deck). Depending on how the finer details are defined, one strategy switch per player may guarantee that no one is better off switching a second time, but if a player is allowed to see that their opponent switched, for example, and that new information incentivizes them to switch again, then no matter how many times players switch while converging to or cycling between strategies, they should not be incentivized to deviate from their superstrategy.
Example #2: Consider a three-player game of Monopoly. I'm not sure if real Monopoly is guaranteed to halt, so assume some arbitrary modifications to the rules to keep things finite, if it matters. Even if finite, the strategy space is huge, but perhaps we think an important factor for predicting outcomes is knowing which players are included in property trades. In real Monopoly, players sometimes collude against the player(s) perceived to be the most skilled at the game by refusing to trade with them. Suppose we design our algorithm to categorize each of Player A's strategies as one of the superstrategies...
- Attempt to collude with Player B against Player C
- Attempt to collude with Player C against Player B
- Do not attempt to collude against either opponent
Player B and Player C would each have three analogous superstrategies. Maybe here we want to evaluate whether Player A colluding against Player C is dominated by colluding against Player B. I can think of at least two senses in which this might be defined...
- Regardless of which strategies Player B and Player C choose, Player A is always better off playing at least one strategy within the collusion against Player B superstrategy than any strategy within the collusion against Player C superstrategy, though knowing the other players' strategies may influence which strategy within the collusion against Player B superstrategy Player A prefers; the invariant is that no strategy within the collusion against Player C superstrategy is ever better
OR
- Regardless of which superstrategies Player B and Player C choose, Player A is always better off in expectation playing at least one strategy within the collusion against Player B superstrategy than any strategy within the collusion against Player C superstrategy, though learning the other players' exact strategies may incentivize Player A to switch to a different superstrategy
Do game theoretic tools extend in this way, possibly with slight modification if I've described them incorrectly? If so, then can qualitative game theoretic conclusions about a game change depending on how finely or coarsely the strategy space is partitioned into superstrategies? If not in general, then how far can this idea be pushed beyond the completely trivial examples before it stops producing correct results?
