 Here the \(x\) and \(y\) axes represent the utilities of Row and Column. The four outcomes entered in the matrix of the second section are represented by the labeled dots. Conditions PD3a and PD3b (see  above)  ensure that \((\bC,\bD)\) and \((\bD,\bC)\) lie northwest and southeast of \((\bD,\bD)\), and PD3c is reflected in the fact that \((\bC,\bC)\) lies northeast of \((\bD,\bD)\). Suppose first that \((\bD,\bD)\) and \((\bC,\bC)\) lie on opposite sides of the line between \((\bC,\bD)\) and \((\bD,\bC)\), as in the graph on the left. Then the four points form a convex quadrilateral, and the payoffs of the feasible outcomes of mixed strategies are represented by all the points on or within this quadrilateral. Of course a player can really only get one of four possible payoffs each time the game is played, but the points in the quadrilateral represent the expected values of the payoffs to the two players. If Row and Column cooperate with probabilities \(p\) and \(q\) (and defect with probabilities \(p^*=1-p\) and \(q^*=1-q\)), for example, then the expected value of the payoff to Row is \(p^*qT+pqR+p^*q^*P+pq^*S\). A rational self-interested player, according to a standard view, should prefer a higher expected payoff to a lower one. In the graph on the left the payoff for universal cooperation (with probability one) is pareto optimal among the payoffs for all mixed strategies. In the graph on the right, however, where both \((\bD, \bD)\) and \((\bC, \bC)\) lie southwest of the line between \((\bC, \bD)\) and \((\bD, \bC)\), the story is more complicated. Here the payoffs of the feasible outcome lie within a figure bounded on the northeast by three distinct curve segments, two linear and one concave. Notice that \((\bC, \bC)\) is now in the interior of the region bounded by solid lines, indicating that there are mixed strategies that provide both players a higher expected payoff than \((\bC, \bC)\). It is important to note that we are talking about independent mixed strategies here. Row and Column use private randomizing devices and have no communication. If they were able to correlate their mixed strategies, so as to ensure, say \((\bC, \bD)\) with probability \(p\) and \((\bD, \bC)\) with probability \(p^*\), the set of feasible solutions would extend up to (and include) the dotted line between \((\bC, \bD)\) and \((\bD, \bC)\). The point here is that, even confined to independent strategies, there are some games satisfying PD3 in which both players can both do better than they do with universal cooperation. A PD in which universal cooperation is pareto optimal may be called a pure PD. (This phenomenon is identified in Kuhn and Moresi and applied to moral philosophy in Kuhn 1996.) A pure PD is characterized by adding to PD3 the following condition.