Game Theory geym theeree a mathematical theory that

Game Theory [geym theer-ee] : a mathematical theory that deals with the general features of competitive situations in a formal abstract way

Definitions • “Two-person game”: game between two people (duh) – Characterized by: • Strategies of Player 1 • Strategies of Player 2 • The payoff table of the players strategies • Zero sum: sum of the net winnings is zero (this property allows payoff tables to be un -unique)

Scenario: Presidential Election • Obama vs. Colbert • Can meet with people at two different times, 9: 00 a. m. or 4: 00 p. m. • Depending on when they go meet, they can gain or lose votes towards the Presidential Election (I understand they’re both democrats, but this is for illustrative purposes)

“Strategy” • Definition: predetermined rule that specifies completely how a player will respond to each possible circumstance at each stage of the game (was going to do chess, way too much strategy involved) • From the two options of time, the opponents can see the strategies involved in how they will gain votes • Strategy 1: Go at 9: 00 a. m. • Strategy 2: Go at 4: 00 p. m. • Due to polling of attendees of the meetings, the suggested results of attendance are demonstrated in a “payoff table”

“Payoff Table” Obama Colbert Strategy 1 2 1 -6 -1 2 4 2 • The strategies reflect the amount of votes won or lost (in hundreds) • This payoff table is specifically for candidate Colbert

“Dominated Strategy” Obama Colbert Strategy 1 2 1 -6 -1 2 4 2 • Dominated Strategy: a strategy that is always at least as good as another strategy (or better) regardless of the opponents strategy • The dominated strategy for Colbert is #2

Omniscient Obama • Old college nickname • Is knowledgeable of all of Colbert’s options (he is a “rational player”) • Having this knowledge he will undoubtedly choose the option that will minimize his losses (or Colbert’s wins)

Choosing of Strategies Obama Colbert Strategy 1 2 1 -6 -1 2 4 2 • Colbert will choose strategy 2 because it is his “dominated strategy” • Obama has no dominated strategy, thus chooses #2 to minimize his losses

Update on Meetings Obama Strategy 1 2 3 1 -3 -2 6 2 2 0 2 3 5 -2 -4 Colbert • The candidates have option 3 now, which is to attend both meeting times to try and get more votes. • No dominated strategy presents itself

Maximin and Minimax Obama Strategy 1 2 3 1 -3 -2 6 2 2 0 2 3 5 -2 -4 Colbert • Maximin: (Colbert) the strategy that will cause the least loss, or where minimum payoff is maximized (play conservatively); this is #2. • Minimax: (Obama) the strategy that will cause the least loss, or where the maximum payoff is minimized; this is #2.

Saddle Point • When the maximin and the minimax values lay on the same point in the payoff table, this creates a “saddle point” • A saddle point is called a stable solution (or an equilibrium solution)

Unstable Universe Obama Strategy 1 2 3 1 0 -2 2 2 5 4 -3 3 2 3 -4 Colbert • The maximin is -2. • The minimax is 2. • Since they do not coincide, there is no saddle point • Unstable Solution!! • Very much like chess…

Introducing… Probability! • Probabilities have to add up to one • xi = probability that Colbert will use strategy i (i=1…m) • yj = probability that Obamaa will use strategy j (j=1…n) • xi and yj are called “mixed strategies” while a non -probabilistic strategy (the ones listed previously) are called “pure strategies” • Mixed strategies lead to pure strategies

Semi-Example Obama Strategy 1 2 3 1 0 -2 2 2 5 4 -3 3 2 3 -4 Colbert • Mixed strategies of (x 1, x 2, x 3) = (0. 5, 0), because sum of xi’s equals 1, (y 1, y 2, y 3) = (0, 0. 5). • This means that Colbert is giving a 50 -50 chance between pure strategies 1 and 2, with no chance of strategy 3; and similarly for Obama.

Expected Payoff • Expected payoff for Colbert: – p is the payoff of the i, j position of the payoff matrix – x and y are the mixed strategies – So, xiyj = 0. 5 x 0. 5 every time, and the payoff is (-2, 2, 4, -3) – Therefore the expected payoff is 0. 25 because (0. 25)*(-2+2+4+-3) = 0. 25

Rerun Maximin and Minimax • Colbert: maximize the minimum payoff (maximin value denoted by v) • Obama: minimize the maximum loss (minimax denoted by ¤, not the recognized symbol)

Maximin Theorem • If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimax criterion provides a stable solution with v = ¤ = v (the value of the game), so that neither player can do better by unilaterally changing her or his strategy. • Value of the game: the payoff to player 1 (Colbert) when both players play optimally • Therefore stable solution is v = ¤, and an unstable solution is where v < ¤

Graphical Solution and Representation • Easiest graphical solutions are two dimensional (like the first example) with the two dimensions being (x 1, x 2)

Dominated Strategy Obama Strategy 1 2 3 1 0 -2 2 2 5 4 -3 3 2 3 -4 Colbert Strategy 1 2 3 1 0 -2 2 2 5 4 -3 Colbert

• Remember x 1 is the mixed strategy for strategy 1 and x 2 is the mixed strategy for strategy 2 • x 2=1 -x 1, so must only solve for x 1 • Can use the graph to evaluate the maximin expected payoff (more votes for Colbert) • Can use the graph to evaluate the minimax mixed strategy

Determine Value of Game • Since it is impossible to determine another players mixed strategy, just use the pure strategies as a starting point to gain some knowledge about the value of the game • For (y 1, y 2, y 3)=(1, 0, 0), and the payoff table of the dominated strategy, the expected payoff becomes 0*x 1 + 5(1 -x 1). Obama Colbert Strategy 1 2 3 1 0 -2 2 2 5 4 -3

Graph • Similar expected payoffs arise from (0, 1, 0) and (0, 0, 1)… lead to graph…

Stable Solution • Using the minimax theorem, solve for the maximin: max{min{-3+5 x 1, 4 -6 x 1}}, the two graphs who intersect at the bottom create the minimax value • Solving for x 1 by setting the two equations equal, we get the optimal mixed strategy for Colbert which is (x 1, x 2)=(7/11, 4/11) and the value of the game is: v = -3 + 5(7/11) = 2/11 • Remember: value of game is the expected payoff for Colbert if both players are playing optimally

• The optimal mixed strategy (7/11, 4/11) simply means that Colbert should more or less choose strategy 1 more times than strategy 2. Obama Colbert Strategy 1 2 3 1 0 -2 2 2 5 4 -3

What about Obama? • Can use similar graphing techniques to determine the minimax for Obama; however, that is time consuming and not conducive to the final linear programming outcome

Linear Programming • Constraints: all mixed strategies (xi thru xm) must add to 1 and nonnegativity • Do not know the value of the game (v) and have no objective function • Make the value of the game the objective function and set it equal to xm+1 • Additional constraints: the set of inequalities from the expected payoff equation, where y=1 at some j and 0 at the rest

Determine Value of Game • (y 1, y 2, y 3)=(1, 0, 0), the expected payoff becomes 0*x 1 + 5(1 x 1). Obama Colbert Strategy 1 2 3 1 0 -2 2 2 5 4 -3

Example Layout • • • Maximize: xm+1 Subject to: p 11 x 1 + p 21 x 2 +…+pm 1 xm - xm+1 >=0 Etc… up until j=n And nonnegativity And mixed strategies sum to 1.

Primal and Dual • Since the payoff table is not player 2’s, his is obviously just a byproduct of player 1’s payoff table, or in OR terms, his is the dual to player 1’s primal • From either the primal or the dual you can solve to obtain the minimax and optimal mixed strategy • But that’s another topic because I didn’t do Dual Theory…