Editing Minimax (section)

=== In general games ===
The '''maximin value''' is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is:<ref name=ZMS2013>{{cite book |author1=Maschler, Michael |author2=Solan, Eilon |author2-link=Eilon Solan |author3=Zamir, Shmuel |year=2013 |title=Game Theory |publisher=[[Cambridge University Press]] |isbn=9781107005488 |pages=176–180}}</ref>

:<math>\underline{v_i} = \max_{a_i} \min_{a_{-i}} {v_i(a_i,a_{-i})}</math>

Where:

* {{mvar|i}} is the index of the player of interest.
* <math>-i</math> denotes all other players except player {{mvar|i}}.
* <math>a_i</math> is the action taken by player {{mvar|i}}.
* <math>a_{-i}</math> denotes the actions taken by all other players.
* <math>v_i</math> is the value function of player {{mvar|i}}.

Calculating the maximin value of a player is done in a worst-case approach: for each possible action of the player, we check all possible actions of the other players and determine the worst possible combination of actions – the one that gives player {{mvar|i}} the smallest value. Then, we determine which action player {{mvar|i}} can take in order to make sure that this smallest value is the highest possible.

For example, consider the following game for two players, where the first player ("row player") may choose any of three moves, labelled {{mvar|T}}, {{mvar|M}}, or {{mvar|B}}, and the second player ("column player") may choose either of two moves, {{mvar|L}} or {{mvar|R}}. The result of the combination of both moves is expressed in a payoff table:

:<math>\begin{array}{c|cc}
\hline
&
L &
R \\
\hline
T &
3,1 &
2,-20
\\
M &
5,0 &
-10,1
\\
B &
-100,2 &
4,4 \\
\hline
\end{array}</math>
(where the first number in each of the cell is the pay-out of the row player and the second number is the pay-out of the column player).

For the sake of example, we consider only [[Strategy (game theory)#Pure and mixed strategies|pure strategies]]. Check each player in turn:
* The row player can play {{mvar|T}}, which guarantees them a payoff of at least {{val|2}} (playing {{mvar|B}} is risky since it can lead to payoff {{val|-100}}, and playing {{mvar|M}} can result in a payoff of {{val|-10}}). Hence: <math>\underline{v_{row}} = 2</math>.
* The column player can play {{mvar|L}} and secure a payoff of at least {{val|0}} (playing {{mvar|R}} puts them in the risk of getting <math>-20</math>). Hence: <math>\underline{v_{col}} = 0</math>.

If both players play their respective maximin strategies <math>(T,L)</math>, the payoff vector is <math>(3,1)</math>.

The '''minimax value''' of a player is the smallest value that the other players can force the player to receive, without knowing the player's actions; equivalently, it is the largest value the player can be sure to get when they ''know'' the actions of the other players. Its formal definition is:<ref name=ZMS2013/>

:<math>\overline{v_i} = \min_{a_{-i}} \max_{a_i} {v_i(a_i,a_{-i})}</math>

The definition is very similar to that of the maximin value – only the order of the maximum and minimum operators is inverse. In the above example:
* The row player can get a maximum value of {{mvar|4}} (if the other player plays {{mvar|R}}) or {{val|5}} (if the other player plays {{mvar|L}}), so: <math>\overline{v_{row}} = 4\ .</math>
* The column player can get a maximum value of {{mvar|1}} (if the other player plays {{mvar|T}}), {{val|1}} (if {{mvar|M}}) or {{val|4}} (if {{mvar|B}}). Hence: <math>\overline{v_{col}} = 1\ .</math>

For every player {{mvar|i}}, the maximin is at most the minimax:
:<math>\underline{v_i} \leq \overline{v_i}</math>
Intuitively, in maximin the maximization comes after the minimization, so player {{mvar|i}} tries to maximize their value before knowing what the others will do; in minimax the maximization comes before the minimization, so player {{mvar|i}} is in a much better position – they maximize their value knowing what the others did.

Another way to understand the ''notation'' is by reading from right to left: When we write
:<math>\overline{v_i} = \min_{a_{-i}} \max_{a_i} {v_i(a_i,a_{-i})} = \min_{a_{-i}} \Big( \max_{a_i} {v_i(a_i,a_{-i})} \Big) </math>
the initial set of outcomes <math>\ v_i(a_i,a_{-i})\ </math> depends on both <math>\ {a_{i}}\ </math> and <math>\ {a_{-i}}\ .</math> We first ''marginalize away'' <math>{a_{i}}</math> from <math>v_i(a_i,a_{-i})</math>, by maximizing over <math>\ {a_{i}}\ </math> (for every possible value of <math>{a_{-i}}</math>) to yield a set of marginal outcomes <math>\ v'_i(a_{-i})\,,</math> which depends only on <math>\ {a_{-i}}\ .</math> We then minimize over <math>\ {a_{-i}}\ </math> over these outcomes. (Conversely for maximin.)

Although it is always the case that <math>\ \underline{v_{row}} \leq \overline{v_{row}}\ </math> and <math>\ \underline{v_{col}} \leq \overline{v_{col}}\,,</math> the payoff vector resulting from both players playing their minimax strategies, <math>\ (2,-20)\ </math> in the case of <math>\ (T,R)\ </math> or <math>(-10,1)</math> in the case of <math>\ (M,R)\,,</math> cannot similarly be ranked against the payoff vector <math>\ (3,1)\ </math> resulting from both players playing their maximin strategy.