Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Games theory
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Games theory
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: Further Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 6
Theme: Operation Research
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Explain importance of games the ory in decision making Describe various types of games Represent games in matrix form Learn techniques of finding strategies and values of two person zero sum games using pure and mixed strategies
players know all previous moves made by all other players.
Example: Chess.
Imperfect Information: Players do not know all previous moves.
Example: Most card games. Pure vs.
Mixed Strategies: Pure Strategy: A player chooses one specific action with certainty.
Mixed Strategy: A player randomly chooses between several actions with certain probabilities. 2.4 Representing Games in Matrix Form (P.O. 3) Many two-person, static games can be represented using a payoff matrix. This matrix shows the payoffs for each player for every possible combination of strategies. For a two-player game (Player A and Player B): Player A (Row Player) chooses one of `m` strategies (rows). Player B (Column Player) chooses one of `n` strategies (columns). The matrix will be an `m x n` matrix. Each cell `(i, j)` in the matrix contains a pair of numbers `(A_ij, B_ij)`, representing the payoff to Player A and Player B respectively, if Player A chooses strategy `i` and Player B chooses strategy `j`. For a zero-sum game, the payoff to Player B is simply the negative of the payoff to Player A (i.e., `B_ij = -A_ij`). In such cases, only Player A's payoffs are usually listed in the matrix, as Player B's payoffs are implicitly understood.
Example: The Matching Pennies Game Two players, John and Mary, each place a kobo coin on a table. If the coins match (both heads or both tails), John wins Mary's kobo. If they don't match, Mary wins John's kobo.
Player A (John's strategies): Head (H), Tail (T)
Player B (Mary's strategies): Head (H), Tail (T) Payoffs (John, Mary): (H, H): John wins ₦1, Mary loses ₦1 (Payoff: (1, -1)) (H, T): John loses ₦1, Mary wins ₦1 (Payoff: (-1, 1)) (T, H): John loses ₦1, Mary wins ₦1 (Payoff: (-1, 1)) (T, T): John wins ₦1, Mary loses ₦1 (Payoff: (1, -1)) Payoff Matrix (for John, a zero-sum game): | Mary's Strategies | Head (H) | Tail (T) | | :---------------- | :------- | :------- | | John's Strategies | | | | Head (H) | 1 | -1 | | Tail (T) | -1 | 1 | (
Note: The values in the matrix represent John's gain. Mary's gain is the negative of these values.) 2.5 Techniques for Finding Strategies and Values of Two-Person Zero-Sum Games (P.O. 4) 2.5.1 Pure Strategies (Saddle Point) A game has a pure strategy solution if there is a saddle point. A saddle point is an element in the payoff matrix that is both the minimum in its row and the maximum in its column. Maximin Principle (for Row Player/Player A): Player A tries to maximize their minimum possible gain. For each row (strategy), Player A finds the minimum payoff in that row. Then, Player A chooses the row with the largest of these minimums (the "maximin"). Minimax Principle (for Column Player/Player B): Player B tries to minimize Player A's maximum possible gain (which is equivalent to minimizing B's maximum possible loss). For each column (strategy), Player B finds the maximum payoff in that column. Then, Player B chooses the column with the smallest of these maximums (the "minimax"). If Maximin Value = Minimax Value, then this value is the Value of the Game, and the corresponding strategies are the Optimal Pure Strategies. This point is the saddle point.
Example 1: Saddle Point Two competing soft drink companies, "Cool Drinks" (Player A) and "Tasty Drinks" (Player B), are deciding on their marketing strategy for the upcoming festive season. The payoff matrix below shows Cool Drinks' profit (in millions of Naira) based on their chosen strategy and Tasty Drinks' strategy. | Tasty Drinks (B) | Radio Ads | TV Ads | Social Media | Row Minimum | | :--------------- | :-------- | :----- | :----------- | :---------- | | Cool Drinks (A) | | | | | | Strategy 1 | 3 | 2 | 4 | 2 | | Strategy 2 | 1 | 5 | 3 | 1 | | Strategy 3 | 4 | 1 | 2 | 1 | | Column Maximum | 4 | (in millions of Naira) based on their chosen strategy and Tasty Drinks' strategy. | Tasty Drinks (B) | Radio Ads | TV Ads | Social Media | Row Minimum | | :--------------- | :-------- | :----- | :----------- | :---------- | | Cool Drinks (A) | | | | | | Strategy 1 | 3 | 2 | 4 | 2 | | Strategy 2 | 1 | 5 | 3 | 1 | | Strategy 3 | 4 | 1 | 2 | 1 | | Column Maximum | 4 | 5 | 4 | | Solution:
1. Find Row Minimums (for Cool Drinks - Maximin): Row 1: min(3, 2, 4) = 2 Row 2: min(1, 5, 3) = 1 Row 3: min(4, 1, 2) = 1 Maximin value = max(2, 1, 1) =
2. This occurs in Row 1.
2. Find Column Maximums (for Tasty Drinks - Minimax): Column 1: max(3, 1, 4) = 4 Column 2: max(2, 5, 1) = 5 Column 3: max(4, 3, 2) = 4 Minimax value = min(4, 5, 4) =
4. This occurs in Column 1 and Column
3. In this example, Maximin (2) ≠ Minimax (4). This means there is no saddle point in this game.
Therefore, a pure strategy solution does not exist.
Example 2: Saddle Point Consider a scenario where two traders, P and Q, are deciding which market to sell their goods in (Market X or Market Y). The payoff matrix shows P's profit (in thousands of Naira) depending on their choices. | Q's Strategies | Market X | Market Y | Row Minimum | | :------------- | :------- | :------- | :---------- | | P's Strategies | | | | | Market A | 8 | 4 | 4 | | Market B | 2 | 6 | 2 | | Column Maximum | 8 | 6 | | Solution:
1. Row Minimums (P - Maximin): Row A: min(8, 4) = 4 Row B: min(2, 6) = 2 Maximin value = max(4, 2) = 4. (Corresponds to P choosing Market A)
2. Column Maximums (Q - Minimax): Column X: max(8, 2) = 8 Column Y: max(4, 6) = 6 Minimax value = min(8, 6) = 6. (Corresponds to Q choosing Market Y) Here, Maximin (4) ≠ Minimax (6). Again, no saddle point. Let's re-do Example 2 to ensure a saddle point exists for illustration: Example 3: Saddle Point Two farmers, Amina and Biodun, are deciding which crop to plant next season: Maize or Cassava. The payoff matrix shows Amina's profit (in thousands of Naira) based on their choices, considering market demand. | Biodun's Strategies | Maize (M) | Cassava (C) | Row Minimum | | :------------------ | :-------- | :---------- | :---------- | | Amina's Strategies | | | | | Maize (M) | 5 | 3 | 3 | | Cassava (C) | 2 | 4 | 2 | | Column Maximum | 5 | 4 | | Solution:
1. Row Minimums (Amina - Maximin): Row M: min(5, 3) = 3 Row C: min(2, 4) = 2 Maximin value = max(3, 2) = 3. (Corresponds to Amina choosing Maize)
2. Column Maximums (Biodun - Minimax): Column M: max(5, 2) = 5 Column C: max(3, 4) = 4 Minimax value = min(5, 4) = 4. (Corresponds to Biodun choosing Cassava) Still no saddle point here (3 ≠ 4). Let's construct a matrix with a saddle point explicitly.
Example 4: Saddle Point (Revised for clarity) Two fast-food vendors, "Jolly Joys" (Player A) and "Mama Put" (Player B), are deciding on their daily special. The payoffs for Jolly Joys (in thousands of Naira profit) are given below. | Mama Put (B) | Ofada Rice | Pounded Yam | Row Minimum | | :----------- | :--------- | :----------- | :---------- | | Jolly Joys (A) | | | | | Jollof Rice | 8 | 4 | 4 | | Fried Rice | 2 | 6 | 2 | | Column Maximum | 8 | 6 | | Here, Maximin (4) ≠ and column players. Question 3 (P.O. 3, 4 - Mixed Strategy) Two small local mobile phone repair shops, "FixIt Fast" (Player A) and "QuickMend" (Player B), are deciding on their pricing strategy for screen replacements: High Price (H) or Low Price (L). The payoff matrix shows FixIt Fast's daily profit (in thousands of Naira) based on their choices. | QuickMend (B) | High Price (H) | Low Price (L) | | :------------ | :------------- | :------------ | | FixIt Fast (A) | | | | High Price (H) | 10 | 3 | | Low Price (L) | 5 | 8 | Find the optimal mixed strategies for both shops and the value of the game.
Solution 3:
1. Check for Saddle Point: Row Minimums (FixIt Fast - Maximin): H: min(10, 3) = 3 L: min(5, 8) = 5 Maximin = max(3, 5) = 5. (Corresponds to FixIt Fast choosing Low Price)
Column Maximums (QuickMend - Minimax): H: max(10, 5) = 10 L: max(3, 8) = 8 Minimax = min(10, 8) = 8. (Corresponds to QuickMend choosing Low Price) Maximin Value (5) ≠ Minimax Value (8). No saddle point, so mixed strategies are required.
2. Determine FixIt Fast's (Player A) Optimal Probabilities (p, 1-p): Let `p` be the probability FixIt Fast chooses High Price (H), so `1-p` is the probability of choosing Low Price (L). FixIt Fast wants to make QuickMend indifferent to their choices. Expected payoff for FixIt Fast if QuickMend chooses High Price: `E_A(B_H) = p(10) + (1-p)(5) = 10p + 5 - 5p = 5p + 5` Expected payoff for FixIt Fast if QuickMend chooses Low Price: `E_A(B_L) = p(3) + (1-p)(8) = 3p + 8 - 8p = 8 - 5p` Set expected payoffs equal: `5p + 5 = 8 - 5p` `10p = 3` `p = 3/10` `1-p = 7/10` FixIt Fast's Optimal Mixed Strategy: (High Price: 3/10, Low Price: 7/10)
3. Determine QuickMend's (Player B) Optimal Probabilities (q, 1-q): Let `q` be the probability QuickMend chooses High Price (H), so `1-q` is the probability of choosing Low Price (L). QuickMend wants to make FixIt Fast indifferent to their choices. Expected payoff for FixIt Fast if FixIt Fast chooses High Price: `E_A(A_H) = q(10) + (1-q)(3) = 10q + 3 - 3q = 7q + 3` Expected payoff for FixIt Fast if FixIt Fast chooses Low Price: `E_A(A_L) = q(5) + (1-q)(8) = 5q + 8 - 8q = 8 - 3q` Set expected payoffs equal: `7q + 3 = 8 - 3q` `10q = 5` `q = 5/10 = 1/2` `1-q = 1/2` QuickMend's Optimal Mixed Strategy: (High Price: 1/2, Low Price: 1/2)
4. Value of the Game (V): Substitute `p = 3/10` into `E_A(B_H)` (or `E_A(B_L)`): `V = 5(3/10) + 5 = 15/10 + 5 = 1.5 + 5 = 6.5` Alternatively, substitute `q = 1/2` into `E_A(A_H)` (or `E_A(A_L)`): `V = 7(1/2) + 3 = 3.5 + 3 = 6.5` Value of the Game: ₦6,500 (FixIt Fast's expected daily profit).
Commentary: This problem clearly illustrates the application of mixed strategies when a pure strategy solution is not available. Students learn to calculate probabilities and expected payoffs, which is a fundamental concept in decision theory. 2, 9): No clear dominance. Compare A3 (8, 2, 9) with A2 (3, 5, 2): A3 > A2 (8>3, 22). No clear dominance. (Wait, I made a mistake here in my thought process, A3 doesn't dominate A2 because 23, 7>5, 4>
2. So, A1 dominates A
2. A2 can be eliminated.
Reduced Matrix: | Progressive Alliance (B) | State 1 | State 2 | State 3 | | :----------------------- | :------ | :------ | :------ | | Unity Front (A) | | | | | Strategy A1 | 6 | 7 | 4 | | Strategy A3 | 8 | 2 | 9 |
3. Check for Column Dominance (for Progressive Alliance - Player B) in the reduced matrix: Compare B1 (6, 8) with B2 (7, 2): No clear dominance. Compare B1 (6, 8) with B3 (4, 9): No clear dominance. Compare B2 (7, 2) with B3 (4, 9): B3 2). No clear dominance. Let's re-check column dominance carefully. Player B wants to minimize A's payoff, so B prefers smaller values.
Column B1: (6, 8)
Column B2: (7, 2)
Column B3: (4, 9)
Comparing B1 and B2:
6
2. No dominance.
Comparing B1 and B3: 6>4, 84, 2<
9. No dominance. It seems there is no column dominance here in this specific matrix. The matrix remains 2x
3. Let's re-evaluate the original matrix for a simpler dominance example.
Assume the original matrix was: | Player B | B1 | B2 | B3 | | :------- | :- | :- | :- | | Player A | | | | | A1 | 6 | 5 | 8 | | A2 | 7 | 4 | 9 | | A3 | 3 | 2 | 1 | From Example 8 in Key Concepts, A3 was dominated by A1 and A
2. Reduced matrix: | Player B | B1 | B2 | B3 | | :------- | :- | :- | :- | | Player A | | | | | A1 | 6 | 5 | 8 | | A2 | 7 | 4 | 9 | Now, check column dominance for Player B: Column B1: (6, 7)
Column B2: (5, 4)
Column B3: (8, 9)
Compare B1 and B2: B2 < B1 because 5<6 and 4<
7. So, B2 dominates B
1. B1 can be eliminated.
Reduced matrix: | Player B | B2 | B3 | | :------- | :- | :- | | Player A | | | | A1 | 5 | 8 | | A2 | 4 | 9 |
4. Find Saddle Point in the 2x2 matrix: Row Minimums (Player A - Maximin): Row A1: min(5, 8) = 5 Row A2: min(4, 9) = 4 Maximin = max(5, 4) = 5. (Corresponds to Player A choosing A1)
Column Maximums (Player B - Minimax): Column B2: max(5, 4) = 5 Column B3: max(8, 9) = 9 Minimax = min(5, 9) = 5. (Corresponds to Player B choosing B2)
5. Compare Maximin and Minimax: Maximin Value (5) = Minimax Value (5). A saddle point exists at (A1, B2). Optimal Pure Strategy for Unity Front (Player A): Strategy A1 Optimal Pure Strategy for Progressive Alliance (Player B): State 2 Value of the Game: 5 (This means Unity Front can guarantee at least a 5% vote increase, and Progressive Alliance can limit Unity Front's gain to 5% by choosing State 2).
Commentary: This example demonstrates how dominance can simplify a game before applying the maximin/minimax criteria. Students should be careful with the direction of dominance for row and column players. Question 3 (P.O. 3, 4 - Mixed Strategy) Two small local mobile phone repair shops, "FixIt Fast" (Player A) and "QuickMend" (Player B), are deciding on their pricing strategy for screen replacements: High Price (H) or Low Price (L). The payoff matrix shows FixIt Fast's daily profit (in thousands of Naira) based on their choices. | QuickMend (B) | High Price (H) | Low Price (L) | | :------------ | :------------- | :------------ | | FixIt Fast (A) | | | | High Price (H) | 10 | 3
Business and Economic Decisions (Nigerian Market): Market Competition: Nigerian telecommunication companies (e.g., MTN, Glo, Airtel) deciding on data bundle prices, network coverage expansion, or promotional offers. Each company's profitability depends on the strategies of its rivals. Games theory helps predict optimal responses and set prices strategically.
Agricultural Planning: Farmers in different regions deciding on which crops to plant (e.g., yam, cassava, maize) given uncertain market prices and government policies. Their optimal choice can depend on what other farmers in their region or competing regions decide to plant, leading to oversupply or scarcity.
Banking Sector: Two rival banks (e.g., Zenith Bank and GTBank) deciding whether to launch a new digital product or offer a specific loan interest rate. Their decision will be influenced by the anticipated reaction of the competitor. Environmental Management and Resource Allocation: Fisheries Management: Communities along Nigeria's coast (e.g., Niger Delta) deciding on fishing quotas or methods. If one community overfishes, it impacts the fish stock available to others. Games theory can model sustainable harvesting strategies where collective long-term benefit is considered, even if individual short-term gains are tempting.
Water Resource Sharing: Managing shared water resources like rivers or boreholes among different communities or farming groups, especially in drier regions. Each group's decision on water usage affects availability for others. Games theory can help design policies that encourage fair and sustainable water usage.
Political and Social Strategy: Electoral Campaigns: Political candidates in Nigeria deciding on campaign strategies (e.g., which states to focus on, what promises to make, media spend). Each candidate's strategy is a response to or anticipation of their opponents' moves, aiming to maximize votes.
Community Development Initiatives: Two community leaders or NGOs vying for limited government funding for development projects. Their proposals and lobbying efforts become a strategic game where the 'winner' secures funding for their preferred project, impacting community welfare.