Several times in McGraw's forum of game theory, I believe, minimax and payoff matrices were mentioned as better forms of 'theorizing' competitive battling. This thread will hopefully elucidate on Minimax applications in general and in battling.
Application in Real Life
Before delving into application in battling, take note that minimax is used in poker, zero-sum games, economics, the stock market fluctuations, and strategic war tactics.
Introduction
Minimax theorem states that in zero-sum (total results add up to zero) games, where one player's gain is the other's player's loss, one player should try to minimize his opponent's maximum payoff, whereas the other player should try to maximize his own minimum payoff. Applied to familiar circumstances, one player should try to minimize the opponent's sweeping oppurtunities or oppurtunities to deal damage, and his opponent should try to deal as much damage as possible.
Supposedly, John von Neumann proved this was an optimal strategy no matter what strategy was used by the opponent. It is unlike prediction, where one is try to beat the other by knowing in advance what the opponent is about to do. Minimax players would take into account ALL possible actions and subsequently choose the action that would account for the LEAST risk. No matter what the opponent does, the minimax player will always mitigate the potential risk. On the other hand, maximin players would always begin to search for the action that would give the highest payoff, considering the minimax player's action.
Assuming both players are aware of each finite possibility and actions, and there are no hidden powers a player can abuse, minimax and maximin are the optimal strategies available and neither player would benefit from changing strategies.
The important thing about minimax theorem is that NO ASSUMPTIONS ARE NECESSARY.
No matter how irrational the opposing player is, the minimax theorem does not care. Probabilities, in minimax's simplest form, does not account for much, or anything at all.
Basic Application
Before stop-loss orders, sell short demands, and a pokemon match, a basic example is here to clarify matters. Take a pie, to be shared among Jack and Jill. Jack states the rules of sharing the pie; 1. Jill can cut it any way possible. 2. Jack may choose the first piece. Jill pondered for some time, and chuckling mischievously, cut the pie in half.
Jill has effectively used minimax to minimize the opponent's maximum payoff, and maximized her own minimum payoff.
A pokemon match!!
Two highly advanced minimax theorem experts are onstage, getting ready for an intense, fight-to-the-death battle. Player 1, in a rage of fury, sends out Tauros. Player 2, meanwhile, sends out the defensive Swampert. As of now, there is something of note:
As of now, gimmicky sets, critical hits and so on will not be regarded. However, they will be factored toward the end. Standard movesets will be assumed (at first).
The ferocious, Choice-Band wielding bull is ready for its first move. He has a total of 5 options which are to 1. Return, 2. Zen Headbutt, 3. Stone Edge, 4. Earthquake, and to 5. switch. As the offensive, Tauros wants to maximize his minimum reward, which is to ram his horns into at least something.
The equally grotesque representative for Player 2, Swampert, is ready for its first move as well. It, too, has 5 options: Surf, Ice Beam, Earthquake, Rest, and to switch. Swampert unfortunately is rather on the defensive as Tauros really does have the capabilities to deal a 2HKO or 3HKO at minimum.
This is where minimax gets tricky. The players do not entirely know each other's team, conflicting with the definition of minimax application. This is no worry, however. The players must merely prepare for the worst and do the best possible. Player 2 must take into account all possible actions that player 1 can possibly do. p2 knows he must switch, otherwise suffering a hardy blow from the potential Return. Gengar seems to be a nice and risky switch that can set up, but no, a Zen Headbutt looms. Aerodactyl seems nice at first, avoiding both Return and Earthquake, but ultimately falls to Stone Edge. Really, switching to Skarmory is his only choice. (Ok, let's assume p2's team is something like: swampert/skarm/gengar/aero/snorlax/starmie)
Player 1, knowing that p2 is using minimax, aims to maximize minimum gain. At this point in time, p1 is quite unaware of p2's potential action due to lack of knowledge of his team. As both players have no definite 'edge' (explained in McGraw's other thread) over another, and both players know each other as highly proficient players, p1 is subsequently aware that p2 is likely to switch to something that resists at least 3 of Tauros' moves. p1 now has two choices: 1. to attack with one of four moves and have a small chance of dealing 'good' damage, but risk losing the offensive 'advantage', or 2. to switch and scout, meanwhile hopefully maintaining the 'edge'. Before continuing let's assume another standard team to p1 (tauros/donphan/gyarados/blissey/raikou/metagross). p1, assuming he decides to switch, might opt for Raikou. Raikou probably is going to fare well against the opponent's switch in, and do decently against Swampert as well even if it does stay in.
This, so far, is an incomplete analysis of Minimax theorem. Therefore, wait for subsequent parts for more detailed, realistic and heuristic approach!
Application in Real Life
Before delving into application in battling, take note that minimax is used in poker, zero-sum games, economics, the stock market fluctuations, and strategic war tactics.
Introduction
Minimax theorem states that in zero-sum (total results add up to zero) games, where one player's gain is the other's player's loss, one player should try to minimize his opponent's maximum payoff, whereas the other player should try to maximize his own minimum payoff. Applied to familiar circumstances, one player should try to minimize the opponent's sweeping oppurtunities or oppurtunities to deal damage, and his opponent should try to deal as much damage as possible.
Supposedly, John von Neumann proved this was an optimal strategy no matter what strategy was used by the opponent. It is unlike prediction, where one is try to beat the other by knowing in advance what the opponent is about to do. Minimax players would take into account ALL possible actions and subsequently choose the action that would account for the LEAST risk. No matter what the opponent does, the minimax player will always mitigate the potential risk. On the other hand, maximin players would always begin to search for the action that would give the highest payoff, considering the minimax player's action.
Assuming both players are aware of each finite possibility and actions, and there are no hidden powers a player can abuse, minimax and maximin are the optimal strategies available and neither player would benefit from changing strategies.
The important thing about minimax theorem is that NO ASSUMPTIONS ARE NECESSARY.
No matter how irrational the opposing player is, the minimax theorem does not care. Probabilities, in minimax's simplest form, does not account for much, or anything at all.
Basic Application
Before stop-loss orders, sell short demands, and a pokemon match, a basic example is here to clarify matters. Take a pie, to be shared among Jack and Jill. Jack states the rules of sharing the pie; 1. Jill can cut it any way possible. 2. Jack may choose the first piece. Jill pondered for some time, and chuckling mischievously, cut the pie in half.
Jill has effectively used minimax to minimize the opponent's maximum payoff, and maximized her own minimum payoff.
A pokemon match!!
Two highly advanced minimax theorem experts are onstage, getting ready for an intense, fight-to-the-death battle. Player 1, in a rage of fury, sends out Tauros. Player 2, meanwhile, sends out the defensive Swampert. As of now, there is something of note:
As of now, gimmicky sets, critical hits and so on will not be regarded. However, they will be factored toward the end. Standard movesets will be assumed (at first).
The ferocious, Choice-Band wielding bull is ready for its first move. He has a total of 5 options which are to 1. Return, 2. Zen Headbutt, 3. Stone Edge, 4. Earthquake, and to 5. switch. As the offensive, Tauros wants to maximize his minimum reward, which is to ram his horns into at least something.
The equally grotesque representative for Player 2, Swampert, is ready for its first move as well. It, too, has 5 options: Surf, Ice Beam, Earthquake, Rest, and to switch. Swampert unfortunately is rather on the defensive as Tauros really does have the capabilities to deal a 2HKO or 3HKO at minimum.
This is where minimax gets tricky. The players do not entirely know each other's team, conflicting with the definition of minimax application. This is no worry, however. The players must merely prepare for the worst and do the best possible. Player 2 must take into account all possible actions that player 1 can possibly do. p2 knows he must switch, otherwise suffering a hardy blow from the potential Return. Gengar seems to be a nice and risky switch that can set up, but no, a Zen Headbutt looms. Aerodactyl seems nice at first, avoiding both Return and Earthquake, but ultimately falls to Stone Edge. Really, switching to Skarmory is his only choice. (Ok, let's assume p2's team is something like: swampert/skarm/gengar/aero/snorlax/starmie)
Player 1, knowing that p2 is using minimax, aims to maximize minimum gain. At this point in time, p1 is quite unaware of p2's potential action due to lack of knowledge of his team. As both players have no definite 'edge' (explained in McGraw's other thread) over another, and both players know each other as highly proficient players, p1 is subsequently aware that p2 is likely to switch to something that resists at least 3 of Tauros' moves. p1 now has two choices: 1. to attack with one of four moves and have a small chance of dealing 'good' damage, but risk losing the offensive 'advantage', or 2. to switch and scout, meanwhile hopefully maintaining the 'edge'. Before continuing let's assume another standard team to p1 (tauros/donphan/gyarados/blissey/raikou/metagross). p1, assuming he decides to switch, might opt for Raikou. Raikou probably is going to fare well against the opponent's switch in, and do decently against Swampert as well even if it does stay in.
This, so far, is an incomplete analysis of Minimax theorem. Therefore, wait for subsequent parts for more detailed, realistic and heuristic approach!
