2x2 Ordinal Games

game	prisoner's dilemma

player 1's ranking	{3,1,4,2}
player 2's ranking	{3,4,1,2}

player 1'schoice	player 2'schoice	outcomeof game	player 1'sranking	player 2'sranking
C	C	CC	3	3
C	N	CN	1	4
N	C	NC	4	1
N	N	NN	2	2

player 2

player 1

	C	N
C	{3,3}	{1,4}
N	{4,1}	{2,2}

The dominant strategy for player 1 is {N}.

The dominant strategy for player 2 is {N}.

The Nash equilibria correspond to outcome(s) {NN}.

A 2×2 ordinal game consists of two players, each with the strategy options of cooperation or noncooperation with the other player (these options are denoted C and N, respectively, in this Demonstration). Each player then ranks the four possible resulting outcomes from four (most desirable for that player) down to 1 (least desirable). The game is summarized in a 2×2 matrix of ordered pairs, with the entries in each ordered pair being the rankings of the corresponding outcome by player 1 and player 2, respectively. A dominant strategy for a player is a strategy (a choice of C or N) with the property that such a choice results in a more favorable outcome for that player than the other choice would, regardless of the other player's choice of strategy. A Nash equilibrium is a choice of strategy by each player with the property that a unilateral change of strategy by either player is unfavorable to that player.

Details

The two most famous examples of 2×2 ordinal games are "The Prisoner's Dilemma", which is shown in the thumbnail, and "Chicken", which is shown in snapshot 1. Snapshot 2 illustrates a game in which one player has a dominant strategy while the other does not, and snapshot 3 illustrates a game in which there is no Nash equilibrium.

For further details, see A. D. Taylor, Mathematics and Politics—Strategy, Voting, Power and Proof, New York: Springer–Verlag, 1995.

External Links

Nash Equilibrium (Wolfram MathWorld)

Permanent Citation

Marc Brodie

"2x2 Ordinal Games" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/2x2OrdinalGames/
Published: March 7, 2011