CMSC 474, Introduction to Game Theory Maxmin and Minmax Strategie Mohammad T. Hajiaghayi Univerity of Maryland
Wort-Cae Expected Utility For agent i, the wort-cae expected utility of a trategy i i the minimum over all poible combination of trategie for the other agent: Example: Battle of the Sexe ( ) min -i u i i, -i Wife trategy w = {(p, Opera), ( p, Football)} Huband trategy h = {(q, Opera), ( q, Football)} u w (p,q) = pq + ( p)( q) = 3pq p q + Huband Wife Opera Football Opera, 0, 0 Football 0, 0, For any fixed p, u w (p,q) i linear in q e.g., if p = ½, then u w (½,q) = ½ q + ½ 0 q, o the min mut be at q = 0 or q = e.g., min q (½ q + ½) i at q = 0 min q u w (p,q) = min (u w (p,0), u w (p,)) = min ( p, p) We can write u w (p,q) intead of u w ( w, h )
Maxmin Strategie Alo called maximin A maxmin trategy for agent i A trategy i that make i wort-cae expected utility a high a poible: Thi in t necearily unique Often it i mixed argmax min u i i, -i i -i ( ) Agent i maxmin value, or ecurity level, i the maxmin trategy wort-cae expected utility: For player it implifie to max i min u i ( i, -i ) -i max min u,
Wife and huband trategie Example w = {(p, Opera), ( p, Football)} h = {(q, Opera), ( q, Football)} Recall that wife wort-cae expected utility i min q u w (p,q) = min ( p, p) Find p that maximize it Max i at p = p, i.e., p = /3 Wife maxmin value i p = /3 Wife maxmin trategy i {(/3, Opera), (/3, Football)} Similarly, Huband maxmin value i /3 min q u w (p,q) Huband Wife Huband maxmin trategy i {(/3, Opera), (/3, Football)} p p Opera p Football Opera, 0, 0 Football 0, 0,
Quetion Why might an agent i want to ue a maxmin trategy?
Anwer Why might an agent i want to ue a maxmin trategy? Ueful if i i cautiou (want to maximize hi/her wort-cae utility) and doen t have any information about the other agent whether they are rational what their payoff are whether they draw their action choice from known ditribution Ueful if i ha reaon to believe that the other agent objective i to minimize i expected utility e.g., -player zero-um game (we dicu thi later in hi eion) Solution concept: maxmin trategy profile all player ue their maxmin trategie
Minmax Strategie (in -Player Game) Alo called minimax Minmax trategy and minmax value Dual of their maxmin counterpart Suppoe agent want to punih agent, regardle of how it affect agent own payoff Agent minmax trategy againt agent A trategy that minimize the expected utility of bet repone to argmin max u, ( ) Agent minmax value i maximum expected utility if agent play hi/her minmax trategy: min max u, ( ) Minmax trategy profile: both player ue their minmax trategie
Example Wife and huband trategie w = {(p, Opera), ( p, Football)} h = {(q, Opera), ( q, Football)} u h (p,q) = pq + ( p)( q) = 3pq p q + Given wife trategy p, huband expected utility i linear in q e.g., if p = ½, then u h (½,q) = ½ q + Max i at q = 0 or q = max q u h (p,q) = ( p, p) Find p that minimize thi Min i at p + = p p = /3 Huband/ minmax value i /3 Wife minmax trategy i {(/3, Opera), (/3, Football)} Huband Wife p Opera Football Opera, 0, 0 Football 0, 0, p p p
Minmax Strategie in n-agent Game In n-agent game (n > ), agent i uually can t minimize agent j payoff by acting unilaterally But uppoe all the agent gang up on agent j Let * j be a mixed-trategy profile that minimize j maximum payoff, i.e., * j arg min max u j j For every agent i j, a minmax trategy for i i i component of -j * j j, j Agent j minmax value i j maximum payoff againt j * * max u j j, - j j ( ) = min We have equality ince we jut replaced j * by it value above - j ( ) max j u j j, - j
Minimax Theorem (von Neumann, 98) Theorem. Let G be any finite two-player zero-um game. For each player i, i expected utility in any Nah equilibrium = i maxmin value = i minmax value In other word, for every Nah equilibrium ( *, *), u ( *, * ) min maxu (, ) max min u (, ) u ( *, * ) Corollary. For two-player zero-um game:{nah equilibria} = {maxmin trategy profile}= {minmax trategy profile} Note that thi i not neceary true for non-zero-um game a we aw for Battle of Sexe in previou lide Terminology: the value (or minmax value) of G i agent minmax value.
Maximin and Minimax via LP Let-u =u = u and let mixed trategie = x = x,, x k and = y = y,, y r, in which player ha k trategie and player ha r trategie. Then u x, y = i j x i y j u i,j = j y j i x i u i,j We want to find x which optimize v = max x min y Since player i doing hi bet repone (in min y i x i u i,j i minimized. Thu v = j i x i y j u i,j = ( j y j ) min j i x i u i,j = min j u(x,y) u(x,y) ) he et y j > 0 only if We have the following LP to find v and the firt player trategy x max v uch that v i x i u i,j for all j i x i = x i 0 i x i u i,j i x i u i,j for any j
Maximin and Minimax via LP Similarly by writing an LP for minimax value v = min y max u(x,y), we x can obtain the econd player trategy min v uch that v j y j u i,j for all i j y j = y j 0 Note that due to Minimax Theorem v = v (v v i trivial jut by definition). Alo (, )= (x, y) i a Nah equilibrium.
Example: Matching Pennie Head Tail Agent trategy: diplay head with probability p Agent trategy: diplay head with probability q u (p, q) = p q + ( p)( q) p( q) q( p) = p q + 4pq u (p, q) = u (p, q) u (p, q) Head,, Tail,, Want to how that {Nah equilibria} = {maxmin trategy profile} = {minmax trategy profile} = {(p = ½, q = ½)} p q
Example: Matching Pennie Head Tail Find Nah equilibria u (p, q) = p q + 4pq u (p, q) = u (p, q) If p = q = ½, then u = u = 0 If agent change to p ½ and agent keep q = ½, then u (p, ½) = p + p = 0 If agent change to q ½ and agent keep p = ½, then u (½, q) = ( q + q) = 0 Thu p = q = ½ i a Nah equilibrium u (p, q) Head,, Tail,, Are there any other? p q
Example: Matching Pennie Head Tail Show there are no other Nah equilibria u (p, q) = p q + 4pq u (p, q) = u (p, q) Conider any trategy profile (p, q) where p ½ or q ½ or both Several different cae, depending on the exact value of p and q In every one of them, either agent can increae u by changing p, or agent can increae u by changing q, or both So there are no other Nah equilibria u (p, q) Head,, Tail,, p q
Example: Matching Pennie Head Tail Find all maxmin trategy profile u (p, q) = p q + 4pq u (p, q) = u (p, q) Head,, Tail,, If agent trategy i p = ½ then regardle of value of q, u (½, q) = q + q = 0 If agent trategy i p > ½ then bet repone i q = 0 (ee the diagram) u (p, 0) = p < 0 If agent trategy i p < ½ then bet repone i q = u (p, ) = + p < 0 Thu ha one maxmin trategy: p = ½ u (p, q) p q Similarly, ha one maxmin trategy: q = ½
Example: Matching Pennie Head Tail Find all minmax trategy profile u (p, q) = p q + 4pq u (p, q) = u (p, q) Head,, Tail,, If agent trategy i p = ½ then regardle of value of q, u (½, q) = ( q + q) = 0 If agent trategy i p > ½ then bet repone i q = 0 (ee the diagram) u (p, 0) = ( p) > 0 If agent trategy i p < ½ then bet repone i q = u (p, ) = ( + p) > 0 Thu ha one minmax trategy: p = ½ u (p, q) p q Similarly, ha one minmax trategy: q = ½