Column
Left Right
Up -10,0 -1,-1
Row
Down -8,-8 0,-10
2_4 Do you recognize Game 2 from your reading or the lecture? You can do a positive linear transformation on the payoffs of a game without changing its essential structure. In fact, you can even do more general positive monotonic transformations, though that will affect the probabilities in a mixed strategy equilibrium. Which game is disguised under the action names of Game 2? For the answer, click here This is again our old friend the Prisoner's Dilemma. Game 2 is really the same as Game 1, just presented a bit differently. The essence of the Prisoner's Dilemma is that the players each have a dominant strategy but they all would be better off if each player failed played a dominated strategy.
A way to very clearly see the difference is to rearrange the game matrix. It should not really matter which column is placed to the left and which to the right. That is just a matter of presentation, and does not change the structure of the game, which depends only on the relation between action combinations and payoffs. (Up, Left) must have a payoff of (-10,0), or we have truly changed the game, but it does not matter that (Up, Left) is in the northwest corner of the matrix. Thus, we can rearrange to get a game which is clearly Game 1 with a couple of actions relabelled:
Column
Right Left
Up -1,-1 -10,0
Row
Down 0,-10 -8,-8
Return to Self Test 1.