102 CHAPTER 3 Moving from Static to Dynamic Games ·························································································· · · · · · · ¯ ¯ ¯ · Let G = ({A } {1,2} , U 1 (.), U 2 (.)) and G = ({A } {1,2} , U 1 (.), U 2 (.)) be two-population · · ¯ · games with finite actions: |A 1 × A 2 | < +. The two-population games G and G are strate- · · gically equivalent if there exist positive constants 1 , and scalars 2 , , such · · · that: · · 1 ¯ 1 2 ¯ 2 ¯ · U 1 (a 1 , a 2 ) = 1 U 1 (a 1 , a 2 ) + 2 U 2 (a 1 , a 2 ) = 1 U 2 (a 1 , a 2 ) + 2 U 1 (a 1 , a 2 ) · · · 1 × A 2 . · for all (a 1 , a 2 ) A · Definition 125: Strategically Equivalent Games The reader can easily verify that a pair strategy equivalent game as defined in Definition 125 is indeed an equivalence relation since it is reflexive, symmetric, and transitive (see Appendix A.1 for more details). ·························································································· · · · · · · · All strategically equivalent two-population games have the same set of equilibrium states. · Proposition 126 ·························································································· · · · · Definition 127