Bielie

When playing Smash purely for fun, items of course are either used or not used depending on personal preference, and stages are similarly banned or not banned based on personal preference.

However, it becomes troublesome for tournament play:

The items-off method, as you pointed out, does have the equilibrium problem on some stages, which unfortunately means that those stages must be banned. However, the items-on method has its own issues:

1. Items, when available, are typically much more powerful than the characters’ built-in attacks, which means that they dominate the built-in attacks. Furthermore, the number of ways to use an item are typically much fewer than the number of ways to use built-in attacks, which means that the items actually dominate more options than they add, reducing the depth of the game. Of course, items are only available part of the time during the match, but during those times the depth is reduced, which is generally frowned upon. (I suppose the overall complexity with items is higher, however.)

2. Items spawn at random times, at random locations, and have random effects. This means that the outcomes of matches between players of similar ability will frequently be determined by randomness from the item spawns. In tournaments, randomized outcomes are of course frowned upon heavily.

So really, what this means is that, for tournaments, both the items-off approach and the items-on approach have significant flaws, but generally players consider the items-off approach to be better overall.

That said, your example is non-zero-sum, and the optimal strategy is to negotiate an alternation of production and defense.

One way to compute an unbeatable strategy in a zero-sum game is simply to compute frequencies of each action such as to have an EV of at least 0 against all the pure strategies - in that case, no mixed strategy can possibly improve outcomes.

Grazie e cordiali saluti,

John Rupert

The 70% rush strategy wins because because winning with rush is a lot more significant than winning with anything else: each time you rush an expanding opponent, you get ahead by 2 (+1 to his -1) points, whereas defending against one only nets you 0.3. So you have to successfully defend his rush seven times to make up for the one time he makes it. Your moves won’t be so perfectly matched to his, so it won’t work.

That said, I’m pretty sure a double-blind game can’t be “solved” in the game theory sense anyway; the best you can do is find an unexploitable strategy. I guess I would (simplistically) define that as a strategy that can only be beat by one that loses to pure randomness.

“in this case, choosing to rush frequently, defend infrequently, and expand rarely.”

If you actually solve that game, isn’t optimal strategy more like defend frequently (70%), rush infrequently (20%), expand rarely (10%)? Rush is so strong that you should weight your choices towards beating it . . .

“In some cases, you can increase complexity by removing choices. ”

Think you meant increase depth by removing choices.