Operations Research · Unit 6
Game Theory
Exam-focused notes for Game Theory (Operations Research, ORS255): what the TU syllabus asks and how it has actually been tested, with 9 solved past questions from this unit.
What this unit covers
- Definition of game in competitive markets
- Assumptions underlying game theory
- Payoff matrices
- Optimal strategies for players
- Game value determination
- Dominance rule method
- Arithmetic method in game theory
- Two-player competitive situations
Optimal strategies for players
Game Theory Problem
Considering this information, answer the question given below.
| Player A's strategy/Player B's strategy | $B_1$ | $B_2$ | $B_3$ | $B_4$ | $B_5$ |
|---|---|---|---|---|---|
| $A_1$ | 20 | 20 | 20 | 120 | 80 |
| $A_2$ | 80 | -20 | -40 | 60 | 60 |
| $A_3$ | -60 | -40 | 20 | 20 | 140 |
| $A_4$ | 120 | 80 | -60 | 60 | 140 |
(a) What would be the optimal strategy for each player?
(b) What is the value of the game?
[5]
Payoff matrix (Player A rows, Player B columns), entries are payoffs to Player A: A\B $B1$ $B2$ $B3$ $B4$ $B5$ ------------------------------ $A1$ 20 20 20 120 80 $A2$ 80 -20 -40 60 60 $A3$ -60 -40 20 20 140 $A4$ 120 80 -60 60 140 This is a $4 \times 5$ two...
Full solved answer →The following is the payoff matrix of a game being played by A and B. Determine the optimal strategies for the players and the value of the game.
$$\begin{array}{|c|c|c|c|}\hline \text{A's strategies} & B_1 & B_2 & B_3 \ \hline A_1 & 120 & -80 & -20 \ A_2 & 60 & 70 & 30 \ A_3 & -100 & 70 & 20 \ \hline \end{array}$$
[5]
Payoff matrix (A's gains): A \ B $B1$ $B2$ $B3$ ------------ $A1$ 120 -80 -20 $A2$ 60 70 30 $A3$ -100 70 20 Row minima (A's guaranteed minimum for each strategy): - $A1$: $\min(120, -80, -20) = -80$ - $A2$: $\min(60, 70, 30) = 30$ - $A3$: $\min(-100, 70, 20...
Full solved answer →Game Theory Problem: Finding Optimal Strategies
Determine the best strategy for each player A and B and the value of the game.
| Player A / Player B | $B_1$ | $B_2$ | $B_3$ | $B_4$ |
|---|---|---|---|---|
| $A_1$ | 40 | 40 | 40 | 40 |
| $A_2$ | 30 | 30 | 20 | 50 |
| $A_3$ | 10 | 30 | 90 | 20 |
[5]
Payoff matrix (Player A gains): A\B B₁ B₂ B₃ B₄ --------------------- A₁ 40 40 40 40 A₂ 30 30 20 50 A₃ 10 30 90 20 This is a 2-person zero-sum game. Strategy Row values Row Min ------------------------------ A₁ 40, 40, 40, 40 40 A₂ 30, 30, 20, 50 20 A₃ 10, ...
Full solved answer →Game Theory Problem: Firm Competition Strategy
Assume that two firms are competing for market share for a particular product. Each firm is considering what promotional strategy to employ for the coming period. Assume that the following payoff matrix describes the increase in market share of Firm A and the decrease in market share for Firm B. Find the optimal strategy for each firm. Also determine value of the game.
$$\begin{array}{|c|c|c|c|}\hline \text{FirmA/FirmB} & \text{No promotion} & \text{Moderate promotion} & \text{Much promotion} \ \hline \text{No promotion} & 5 & 0 & -10 \ \text{Moderate promotion} & 10 & 6 & 2 \ \text{Much promotion} & 20 & 15 & 10 \ \hline \end{array}$$
[5]
Payoff matrix (Firm A's gain = Firm B's loss), rows = Firm A strategies, columns = Firm B strategies: Firm A \ Firm B No promotion Moderate promotion Much promotion ------------ No promotion 5 0 -10 Moderate promotion 10 6 2 Much promotion 20 15 10 This is ...
Full solved answer →Game Theory Problem: NTC vs Ncell
The following table gives the payoff in million in the competitive situation between Nepal Telecom Communication (NTC) and Ncell. Determine the optimal strategies for each of the company and find the game value.
$$\begin{array}{|c|c|c|c|}\hline \text{NTC}\backslash\text{Ncell} & \text{No Adv} & \text{Med Adv} & \text{High Adv} \ \hline \text{No Adv} & 50 & 40 & 28 \ \text{Med Adv} & 70 & 50 & 45 \ \text{High Adv} & 75 & 52 & 50 \ \hline \end{array}$$
[5]
Two-person zero-sum game. NTC = row player (maximizer), Ncell = column player (minimizer). Payoff matrix (NTC's gains, in millions): NTC \ Ncell No Adv Med Adv High Adv ------------ No Adv 50 40 28 Med Adv 70 50 45 High Adv 75 52 50 Strategy No Adv Med Adv ...
Full solved answer →Dominance rule method
Describe the dominance rule of solving game theory problem. [5]
The dominance rule (or principle of dominance) is a method used to simplify and solve game theory problems by eliminating strategies that are clearly inferior for a player, regardless of what the opponent does. A strategy is said to be dominated if there ex...
Full solved answer →Describe dominance rule method for solving a problem of game theory. [5]
The dominance rule method is a technique used to simplify and solve game theory problems by eliminating strategies that are clearly inferior (dominated) for a player, regardless of what the opponent does. A strategy is dominated if there exists another stra...
Full solved answer →Payoff matrices
Game Theory Problem: Telecom Companies ABC and XYZ
The following table gives a payoff in millions in the competitive situation between telecom companies ABC and XYZ. Determine the optimal strategies for each company and find the game value.
$$\begin{array}{|c|ccc|}\hline \text{ABC's Strategies/XYZ's Strategies} & \text{No advertising} & \text{Medium advertising} & \text{High advertising} \ \hline \text{No advertising} & 50 & 40 & 28 \ \hline \text{Medium advertising} & 70 & 50 & 45 \ \hline \text{High advertising} & 75 & 52 & 50 \ \hline \end{array}$$
[5]
Two-person zero-sum game. ABC = row player (maximizer), XYZ = column player (minimizer). Payoff matrix (ABC's gains in millions): ABC \ XYZ No Adv Med Adv High Adv ------------ No advertising 50 40 28 Medium advertising 70 50 45 High advertising 75 52 50 - ...
Full solved answer →Definition of game in competitive markets
What is called a game in a competitive market? State the assumptions underlying in game theory. [5]
A game in a competitive market context refers to a strategic situation where: - Two or more decision-makers (players/firms) interact with each other - Each player's outcome or payoff depends not only on their own decisions but also on the decisions made by ...
Full solved answer →Make Unit 6 stick
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