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” The strategies reflect the amount of votes won or lost (in hundreds) This payoff table is specifically for candidate Colbert Obama Colbert Strategy

“Dominated Strategy” 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 Obama Colbert Strategy

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 Colbert will choose strategy 2 because it is his “dominated strategy” Obama has no dominated strategy, thus chooses #2 to minimize his losses Obama Colbert Strategy

Update on Meetings The candidates have option 3 now, which is to attend both meeting times to try and get more votes. No dominated strategy presents itself Obama Colbert Strategy

Maximin and Minimax 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. Obama Colbert Strategy

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

Introducing… Probability! Probabilities have to add up to one x i = probability that Colbert will use strategy i (i=1…m) y j = probability that Obamaa will use strategy j (j=1…n) x i and y j 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 Mixed strategies of (x 1, x 2, x 3 ) = (0.5,0.5,0), because sum of x i ’s equals 1, (y 1, y 2, y 3 ) = (0,0.5,0.5). This means that Colbert is giving a chance between pure strategies 1 and 2, with no chance of strategy 3; and similarly for Obama. Obama Colbert Strategy

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, x i y j = 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)*( ) = 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 Colbert Strategy Obama Colbert Strategy

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 (y1,y2,y3)=(1,0,0), and the payoff table of the dominated strategy, the expected payoff becomes 0*x1 + 5(1-x1). Obama Colbert Strategy

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+5x1,4-6x1}}, the two graphs who intersect at the bottom create the minimax value Solving for x1 by setting the two equations equal, we get the optimal mixed strategy for Colbert which is (x1,x2)=(7/11,4/11) and the value of the game is: v = (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

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 x m+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 (y1,y2,y3)=(1,0,0), the expected payoff becomes 0*x1 + 5(1- x1). Obama Colbert Strategy

Example Layout Maximize: x m+1 Subject to: p 11 x 1 + p 21 x 2 +…+p m1 x m - x m+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…