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I'm working with a two-player normal-form game represented by the following matrix:

U D
L 2,0 2,0
R 6,1 3,2

I attempted to find the Nash equilibrium in mixed strategies and concluded with the solution to play L with probability 1 and R with probability 0. However, this doesn't align with the observation that R strictly dominates L for player 1.

I'm puzzled by the following:

  1. If my calculations yield a negative probability, does that mean there is no mixed strategy for either player or only for the player I'm checking on?
  2. Since the Nash equilibrium found seems to be (R,D), how can my calculation yield playing L with probability 1?

What am I missing in my analysis? Can someone help me understand how to properly find the Nash equilibrium for this game?

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1 Answer 1

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Probably you did a calculation mistake. The pure NE is indeed $(R,D)$, since $R$ dominates $L$ for the row player. You can also draw the response graph of the game, that looks like this:

enter image description here

The nodes of this graph are the strategy profiles $\mathcal{N} = \{ (LU), (LD), (RU), (RD) \}$, and the edges are unilateral deviations, i.e. two nodes are connected by an edge iff they differ by the strategy of precisely one player (in particular note that there is no edge between $(LD)$ and $(RU)$, and between $(LU)$ and $(RD)$.

Orient the graph by labelling and orienting each edge with the payoff difference of the player who is deviating along the edge, such that this number is non-negative. So

  • the edge on the left is labelled $6-2 = 4$ and oriented downwards, since the row player is gaining 4 by deviating from $L$ to $R$ if the column player plays $U$
  • the edge on the right is labelled $3-2 = 1$ and oriented downwards, since the row player is gaining 1 by deviating from $L$ to $R$ if the column player plays $D$
  • the edge on the top is labelled $2-2 = 0$ and oriented indifferently, since the column player is indifferent between $U$ and $D$ if the row player plays $L$
  • the edge on the bottom is labelled $2-1 = 1$ and oriented towards the right, since the column player is gains 1 by deviating from $U$ to $D$ if the row player plays $R$.

Now by definition if a node has only incoming arrows, then it is a pure NE.

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