Assignment 7 β Game Theory & AHP
Three competitive games turned from sentences into payoff matrices and solved by the saddle-point test, and two AHP problems taken from a pairwise matrix to priority weights, a ranking and a consistency check.
- Game β formulate: the row player is the firm whose "advantage" the payoff measures. Convert every sentence to that firm's gain: its gain +k, its loss βk, the rival's gain βk, the rival's loss +k, equal/random award 0.
- Game β solve: row minima β maximin; column maxima β minimax. Equal βΉ saddle point, pure strategies, value V = that cell. Unequal βΉ reduce by dominance, then the 2 Γ 2 mixed-strategy formula.
- AHP: complete the matrix with reciprocals (if A vs B = k, B vs A = 1/k) β add each column β divide every entry by its column total β average each row = priority weights (they add to 1) β rank.
- Interpret in words: which strategy each firm should use and whether the game is fair (V = 0); which vendor/supplier is preferred and by how much.
Theory: Unit 4 Β· Theory of Games Β· Unit 7 Β· AHP.
- Circle the row minima, box the column maxima β a value both circled and boxed is the saddle point. All three games here have one.
- AHP sanity check: the row with the biggest numbers in the original matrix must get the biggest weight, and the weights must sum to 1.
- Consistency (one optional line that impresses): CR = CI / RI with RI = 0.58 for three items; CR < 0.10 means the judgements are consistent.
Game Q1 β TBL Cargo vs ABL Cargo
Each firm chooses a Low-tariff (L) or Premium-tariff (P) strategy; the payoff is ABL's expected profit advantage (βΉ lakh). Both low: ABL's payoff 1. ABL low, TBL premium: TBL's payoff is 6. ABL premium, TBL low: ABL's payoff 5. Both premium: ABL's payoff 2. Prepare the zero-sum payoff matrix.
Assignment 7 p. 1
- Step 1 β Row player = ABL (the payoff is ABL's advantage) ABL gains βΉ +; TBL gains βΉ β for ABL "TBL's payoff is 6" βΉ ABL's payoff β6
- Step 2 β Payoff matrix with the saddle-point test
ABL β / TBL β TBL Low (L) TBL Premium (P) Row min ABL Low (L) 1 β6 β6 ABL Premium (P) 5 2 β 2 Column max 5 2 - Step 3 β Interpret
Maximin = 2 = Minimax = 2 βΉ saddle point at (ABL P, TBL P)
Value of the game V = 2 (βΉ2 lakh in ABL's favour)
Both should quote the premium tariff. The game is not fair: ABL keeps a βΉ2 lakh advantage whatever TBL does, because ABL's premium row (5, 2) beats its low row (1, β6) in every column.
Game Q2 β Auto Ltd vs Moto Ltd β three pricing strategies
Strategies H (high, written "A" in two sentences), M (moderate), P (premium). Payoff = Auto's profit advantage. H,H: Auto gains 4 Β· H,M: Moto loses 6 Β· H,P: Auto gains 5 Β· M,H: "moderate pricing loses 3" Β· M,M: both gain equally Β· M,P: "premium gains 1" Β· P,H: "premium gains 2" Β· P,M: "premium gains 4" Β· P,P: Auto gains 1.
Assignment 7 p. 1
- Step 1 β Decode each sentence to Auto's payoff (Auto is the first-named, row player)
(Auto, Moto) Sentence Auto's payoff H, H Auto gains 4 +4 H, M Moto loses 6 βΉ Auto gains 6 +6 H, P Auto gains 5 +5 M, H the one playing moderate (Auto) loses 3 β3 M, M equal gains βΉ no advantage 0 M, P the one playing premium (Moto) gains 1 β1 P, H the one playing premium (Auto) gains 2 +2 P, M the one playing premium (Auto) gains 4 +4 P, P Auto gains 1 +1 The wording is loose ("A" for high, "premium gains") β state your reading in one line, as here, before the matrix.
- Step 2 β Payoff matrix
Auto β / Moto β Moto H Moto M Moto P Row min Auto H 4 β 6 5 4 Auto M β3 0 β1 β3 Auto P 2 4 1 1 Column max 4 6 5 - Step 3 β Interpret
Maximin = 4 = Minimax = 4 βΉ saddle point at (H, H)
V = 4: Auto keeps a βΉ4 lakh advantage
Both firms should choose high pricing. Auto's H row dominates its other rows outright, and against Auto's H, Moto loses least by also pricing high.
Game Q3 β Solar Ltd vs Green Ltd β bidding tariffs
Payoff = Green's advantage over Solar. Green L: loses 3, 7, 6 against Solar's L, M, H. Green M: gains 2, 5, 4. Green H: gains 1, 3, 4.
Assignment 7 p. 1
- Step 1 β Row player = Green; losses are negative Green L row: β3, β7, β6 Green M row: 2, 5, 4 Green H row: 1, 3, 4
- Step 2 β Payoff matrix
Green β / Solar β Solar L Solar M Solar H Row min Green L β3 β7 β6 β7 Green M 2 β 5 4 2 Green H 1 3 4 1 Column max 2 5 4 - Step 3 β Interpret
Maximin = 2 = Minimax = 2 βΉ saddle point at (Green M, Solar L)
V = 2 in Green's favour
Green should bid a moderate tariff; Solar's best reply is a low tariff, holding Green's advantage to βΉ2 lakh. Dominance agrees: Green's M row beats its L row everywhere, and Solar's L column is smaller than its M and H columns everywhere.
AHP problems
Both use your lecture's method β normalise each column, average each row Lecture p. 25 β the same method as the 2025-26 answer keys.
AHP Q1 β Choosing a software vendor
| Vendor A | Vendor B | Vendor C | |
|---|---|---|---|
| Vendor A | 1 | 1/3 | 5 |
| Vendor B | 1 | 7 | |
| Vendor C | 1 |
(1) How does Vendor B compare with A and C? (2) Using AHP priorities, which vendor is most and which least preferred?
Assignment 7 p. 1
- Step 1 β Complete with reciprocals, add the columns
Vendor A Vendor B Vendor C Vendor A 1 1/3 = 0.333 5 Vendor B 3 1 7 Vendor C 1/5 = 0.2 1/7 = 0.143 1 Column total 4.2 1.476 13 A vs B = 1/3 means B vs A = 3; A vs C = 5 βΉ C vs A = 1/5; B vs C = 7 βΉ C vs B = 1/7.
- Step 2 β Normalise and average each row
Check: 0.283 + 0.643 + 0.074 = 1.000 βVendor A Vendor B Vendor C Priority Vendor A 0.238 0.226 0.385 0.283 Vendor B 0.714 0.677 0.538 0.643 Vendor C 0.048 0.097 0.077 0.074 - (1) Vendor B against the others
B vs A = 3 β B is moderately more competitive than A. B vs C = 7 β B is very strongly more competitive than C.
- (2) Recommendation
Vendor B most recommended (0.643), then A (0.283); Vendor C least preferred (0.074)
Consistency: Ξ»max = 3.066, CI = 0.033, CR = 0.056 < 0.10 β the CIO's judgements are consistent.
AHP Q2 β Solar equipment suppliers β quality and reliability
| A1 | A2 | A3 | |
|---|---|---|---|
| A1 | 1 | 1/3 | 4 |
| A2 | 1 | 6 | |
| A3 | 1 |
(a) Complete the matrix using the reciprocal property. (b) Calculate the priority weights. (c) Rank the suppliers. (d) Interpret the Plant Head's judgement and recommend a supplier.
Assignment 7 p. 2
- (a) Completed matrix
A1 Alpha A2 Beta A3 Gamma A1 Alpha 1 1/3 = 0.333 4 A2 Beta 3 1 6 A3 Gamma 1/4 = 0.25 1/6 = 0.167 1 Column total 4.25 1.5 11 - (b) Priority weights
Check: 0.274 + 0.639 + 0.087 = 1.000 βA1 Alpha A2 Beta A3 Gamma Priority A1 Alpha 0.235 0.222 0.364 0.274 A2 Beta 0.706 0.667 0.545 0.639 A3 Gamma 0.059 0.111 0.091 0.087 - (c) Ranking A2 Beta (0.639) > A1 Alpha (0.274) > A3 Gamma (0.087)
- (d) Interpretation
The Plant Head rates Beta moderately better than Alpha (3) and strongly-to-very-strongly better than Gamma (6), and Alpha moderately-to-strongly better than Gamma (4). Beta carries almost two-thirds of the total weight, so Supplier Beta (A2) is recommended for equipment quality and reliability. CR = 0.047 < 0.10, so the judgements are consistent.