Thus, to find Nash equilibria in Γ N =[N, {null(S i )}, {u i }], we use the conditions: for each player i, (1) he is indi?erent among all strategies in S + i , and (2) any strategy in S + i is at least as good as any strategy in S 0 i . Example 3.7. (Meeting in an Airport). Mr.Wang and Ms.Yang are to meet in an airport.However,theydonotknowwhethertheyaretomeetatdoorAordoorB.The payo?s are specified in the following normal form game: Ms. Yang A (σ y ) B (1?σ y ) Mr. Wang: A (σ w ) 20, 20 0, 0 B (1?σ w ) 0, 0 10, 10 Find the Nash equilibria.  Proposition 3.2. If a strategy profile (s ? i ,σ ? ?i ) is a NE in game [N, {S i ,null(S ?i )}, {u i }], it must be a NE in [N, {?(S i ),null(S ?i )}, {u i }].  Hence, if a pure strategy is a NE strategy from the pure-strategy set, it must be a NE strategy from the mixed-strategy set. Example 3.8. Given the following normal-form game, P2 L 2 (σ 12 ) R 2 (σ 22 ) P1: L 1 (σ 11 ) 2, 3 -2, 2 R 1 (σ 21 ) -2, 3 4, 5 find the Nash equilibria.  Proposition 3.3. (Existence). If S i ,i=1,...,n, are finite sets, there exists a NE in Γ N =[N, {null(S i )}, {u i }].  3—5 Proposition 3.4. (Existence). A NE exists in Γ N =[N, {S i }, {u i }] if, for all i = 1,...,n, (a) S i is nonempty, convex, and compact subset of some Euclidean space R m . (b) u i (s 1 ,...,s n ) is continuous in (s 1 ,...,s n ) andquasiconcaveineachs i .  2.2. Dominant-Strategy Equilibrium Definition 3.8. Astrategys i ∈S i for player i is strictly dominated in game Γ N = [N,{S i },{u i }] if there exists another strategy s 0 i ∈ S i such that u i (s 0 i ,s ?i ) >u i (s i ,s ?i ) for all s ?i ∈S ?i . In this case, we say that strategy s 0 i strictly dominates strategy s i . A strategy s i is a strictly dominant strategy for player i in game Γ N =[N,{S i },{u i }] if it strictly dominates every other strategy in S i . Astrategys i ∈S i for player i is weakly dominated if there exists another strategy s 0 i ∈S i such that u i (s 0 i ,s ?i ) ≥ u i (s i ,s ?i ) for all s ?i ∈S ?i and with strict inequality for some s ?i .  Definition 3.9. Astrategyσ i ∈null(S i ) for player i is strictly dominated in game Γ N =[N,{null(S i )},{u i }] if there exists another strategy σ 0 i ∈null(S i ) such that u i (σ 0 i ,σ ?i ) >u i (σ i ,σ ?i ) for all σ ?i ∈ T j6=i null(S j ). In this case, we say that strategy σ 0 i strictly dominates strategy σ i . Astrategyσ i is a strictly dominant strategy for player i in game Γ N =[N,{null(S i )},{u i }] if it strictly dominates every other strategy in null(S i ). Weak dominance is similarly defined.  Proposition 3.5. Player i’s strategy σ 0 i ∈null(S i ) strictly dominates σ i in Γ N = [N,{null(S i )},{u i }] i? u i (σ 0 i ,s ?i ) >u i (σ i ,s ?i ) for all s ?i ∈S ?i .  Thus, to determine dominance of σ 0 i over σ i , we need only compare them against the pure strategies of player i’s opponents. Proposition 3.6. For player i, if a pure strategy ˉs i is strictly dominated by a mixed strategy that assigns zero probability to ˉs i , then every mixed strategy that assigns a positive probability to ˉs i is strictly dominated by a mixed strategy that assigns zero probability to ˉs i .  Thus, when trying to find Nash equilibria, we can iteratively eliminate strictly domi- nated strategies. The order of elimination doesn’t matter. 3—6 Example 3.9. Find all the pure-strategy Nash equilibria in the following game: P2 L 2 M 2 R 2 P1: L 1 2, 3 -2, 2 5, 2 M 1 -2, 3 4, 5 2, 3 R 1 1, 4 -3, -1 8, 1 3—7