Thus, to find Nash equilibria in IN=N, A(Si)), uin, we use the conditions: for ach player i (1)he is indifferent among all strategies in St,and (2)any strategy in St is at least as good as any strategy in S Example 3.7.(Meeting in an Airport ). Mr Wang and Ms Yang are to meet in an irport. However, they do not know whether they are to meet at door a or door B. The payoffs are specified in the following normal form game Ms. Yang A(ou) B(1-Ov Mr. Wang: A(ow) 20.20 B 0.0 10.10 Find the Nash equilibria. L Proposition 3.2. If a strategy profile(si, a*i) is a NE in game N, Si, A(s-i), uinl it must be a ne in,{△),△(s-)},{u 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 L2(o12)R2(o2) P1:L1(o1)2.3 R1(2)-2,34,5 find the Nash equilibria. Proposition 3.3.(Existence). If Si, i=1,., n, are finite sets, there exists a NE rN=N,{△(s)},{u}
Thus, to find Nash equilibria in ΓN = [N, {7(Si)}, {ui}], we use the conditions: for each player i, (1) he is indifferent among all strategies in S+ i , and (2) any strategy in S+ i is at least as good as any strategy in S0 i . Example 3.7. (Meeting in an Airport). Mr.Wang and Ms. Yang are to meet in an airport. However, they do not know whether they are to meet at door A or door B. The payoffs are specified in the following normal form game: Ms. Yang A (σy) B (1 − σy) Mr. Wang: A (σw) 20, 20 0, 0 B (1 − σw) 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, {Si, 7(S−i)}, {ui}], it must be a NE in [N, {∆(Si), 7(S−i)}, {ui}]. 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 L2 (σ12) R2 (σ22) P1: L1 (σ11) 2, 3 -2, 2 R1 (σ21) -2, 3 4, 5 find the Nash equilibria. Proposition 3.3. (Existence). If Si, i = 1, . . . , n, are finite sets, there exists a NE in ΓN = [N, {7(Si)}, {ui}]. 3—5
Proposition 3.4.(Existence). A NE exists in TN=N, S, uil if, for all (a)S: is nonempty, convex, and compact subset of some Euclidean space Rm (b )ui(sl,., sn) is continuous in(s1,., Sn) and quasiconcave in each si 2.2. Dominant-Strategy Equilibrium Definition 3.8. A strategy si E Si for player i is strictly dominated in game IN N, Si, uin if there exists another strategy s;E S: such that u (s, s-i)>ui(si, s_i) for all s_i ES_i. In this case, we say that strategy s strictly dominates strategy Si. A strategy si is a strictly dominant strategy for player i in game TN=N, S, uin if it strictly dominates every other strategy in Si. A strategy Si E Si for player i is weakly dominated if there exists another strategy sE Si such that ui (ss-i2ui(si, si) for all s_iE S_i and with strict inequality for some si. Definition 3.9. A strategy oi E A(Si) for player i is strictly dominated in gan N=(N,{△(S)},{u} if there exists another strategy o∈△(s) such that u4(01,0-)>a(;,O-) for all a-t∈I1≠△(③). In this case, we say that strategy o, strictly dominates strategy oi. A strategy gi is a strictly dominant strategy for player i in game TN=N, A(Si), uin if it strictly dominates every other strategy in △(s2). Weak dominance is similarly defined.■ Proposition 3.5. Player i's strategy a E A(Si) strictly dominates ai in IN N,{△(S)},{u动it(o1,s-)>tl(01,S-) for all s-:∈S- Thus, to determine dominance of a over oi, we need only compare them against the pure strategies of player i's opponents Proposition 3.6. For player i, if a pure strategy S: is strictly dominated by a mixed strategy that assigns zero probability to si, then every mixed strategy that assigns a positive probability to si is strictly dominated by a mixed strategy that assigns zero probability to s Thus, when trying to find Nash equilibria, we can iteratively eliminate strictly domi- nated strategies. The order of elimination doesn't matter
Proposition 3.4. (Existence). A NE exists in ΓN = [N, {Si}, {ui}] if, for all i = 1, . . . , n, (a) Si is nonempty, convex, and compact subset of some Euclidean space Rm. (b) ui(s1,...,sn) is continuous in (s1,...,sn) and quasiconcave in each si. 2.2. Dominant-Strategy Equilibrium Definition 3.8. A strategy si ∈ Si for player i is strictly dominated in game ΓN = [N, {Si}, {ui}] if there exists another strategy s0 i ∈ Si such that ui(s0 i, s−i) > ui(si, s−i) for all s−i ∈ S−i. In this case, we say that strategy s0 i strictly dominates strategy si. A strategy si is a strictly dominant strategy for player i in game ΓN = [N, {Si}, {ui}] if it strictly dominates every other strategy in Si. A strategy si ∈ Si for player i is weakly dominated if there exists another strategy s0 i ∈ Si such that ui(s0 i, s−i) ≥ ui(si, s−i) for all s−i ∈ S−i and with strict inequality for some s−i. Definition 3.9. A strategy σi ∈ 7(Si) for player i is strictly dominated in game ΓN = [N, {7(Si)}, {ui}] if there exists another strategy σ0 i ∈ 7(Si) such that ui(σ0 i, σ−i) > ui(σi, σ−i) for all σ−i ∈ T j6=i 7(Sj ). In this case, we say that strategy σ0 i strictly dominates strategy σi. A strategy σi is a strictly dominant strategy for player i in game ΓN = [N, {7(Si)}, {ui}] if it strictly dominates every other strategy in 7(Si). Weak dominance is similarly defined. Proposition 3.5. Player i’s strategy σ0 i ∈ 7(Si) strictly dominates σi in ΓN = [N, {7(Si)}, {ui}] iff ui(σ0 i, s−i) > ui(σi, s−i) for all s−i ∈ S−i. Thus, to determine dominance of σ0 i over σi, we need only compare them against the pure strategies of player i’s opponents. Proposition 3.6. For player i, if a pure strategy s¯i is strictly dominated by a mixed strategy that assigns zero probability to s¯i, then every mixed strategy that assigns a positive probability to s¯i is strictly dominated by a mixed strategy that assigns zero probability to s¯i. Thus, when trying to find Nash equilibria, we can iteratively eliminate strictly dominated strategies. The order of elimination doesn’t matter. 3—6
Example 3.9. Find all the pure-strategy Nash equilibria in the following game L2 M. R2 M1-2,34,52,3 R11,4-3.-18,1
Example 3.9. Find all the pure-strategy Nash equilibria in the following game: P2 L2 M2 R2 P1: L1 2, 3 -2, 2 5, 2 M1 -2, 3 4, 5 2, 3 R1 1, 4 -3, -1 8, 1 3—7
