[Math] The Notion Of Degenerate Two Player Game

game theory

I try to get the intuitive understanding of the notion "degenerate two player game".

Definition. A two-player game is called non degenerate if no mixed strategy
of support size $k$ has more than $k$ pure best responses.

What's so special about degenerate game, why sometimes we should consider degenerate case and non degenerate case separately?

Whether the definition of degenerate game is applied to non two player game(so far I saw only 2 player version)?

Consider the following two player game in standard form.

$0/0 \space \space \space \space \space 10/10$

$0/0 \space \space \space \space \space 10/10$

When the column player picks pure strategy to play column two. The row player has more than one pure best response: the first row and the second row.

Does it mean that this always happens in degenerate game, that we have few solutions with the same payoffs for each player? On the other hand, in the above example, we can just use mixed strategy for the row player: play the first row with probability 0.5 and play the second row with probability 0.5.

As you see I really confused by the definition of degenerate game and will appreciate any help.

Best Answer

Does it mean that this always happens in degenerate game, that we have few solutions with the same payoffs for each player?

Not few. Infinite. In the example row player can play infinite mixed strategies which are best response to the column player strategy. And yes, this is a property of all degenerate games.

On the other hand, in the above example, we can just use mixed strategy for the row player: play the first row with probability 0.5 and play the second row with probability 0.5.

If you choose that mixed strategy the game remains degenerate. In fact the dual definition says that a 2-player game is degenerate if exists a strategy of support k for which the number of pure best responses of the adversary are more than k.

The definition can be applied also in games with more than two players. If, in a n-players game, exists a strategy s of n-1 players, such that the n-th player has a number of pure best responses greater than the size of the smallest support in the strategy s, then the game is degenerate. One example follows:

     A                 B
0/0/0 0/10/10     0/0/0 0/10/10
0/0/0 0/10/10     0/0/0 0/10/10

The first player (player 1) can choose which table (A or B) to use, while other two players are row player and column player. If player 1 chooses a strategy of (0.5,0.5) and column player chooses the second column (support 1), then the row player is indifferent to play first or second row (2 best responses). This game is degenerate because Nash equilibria are not isolated points.

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