# Game Theory Concept Card: [Your strategic situation, course or case] > A concept card for the game theory taught in the MBA core microeconomics and strategy courses: normal and extensive > form, dominant strategies, Nash equilibrium, mixed strategies, repeated games and cooperation, sequential games and > credible commitment, and the basics of auctions, each with a fully computed business example. It is for MBA students > preparing cases and exams, and for managers who must anticipate how a rival, a supplier or a bidder will respond. > Source basis: von Neumann and Morgenstern (1944), Nash (1950, 1951), Schelling (1960), Vickrey (1961), Selten (1965, > 1975), Friedman (1971), Axelrod (1984) and Myerson (1981), written up in original words with fictional examples. > Replace every [bracketed] value and every row marked *Example*. This is a learning aid, not legal or competition-law > advice. ## Document control | Field | Value | |---|---| | Learner | [Your name] | | Course and session | [Microeconomics or strategy, sessions 8 to 10] | | Situation modelled | [Example: price war risk with a regional rival] | | Source of payoff estimates | [Case exhibits, own estimates, finance team model, YYYY-MM-DD] | | Spreadsheet used | [Excel / Google Sheets / LibreOffice Calc] | | Reviewed by | [Study partner, teaching assistant or mentor] | | Version | [0.1] | | Last updated | YYYY-MM-DD | ## Contents - When to use it - The idea in plain language - Formulas and notation - Normal form, dominance and Nash equilibrium - Mixed strategies - Repeated games and cooperation - Sequential games and credible commitment - Auctions basics - Assumptions and limits - How managers use it - Worked example A (fully solved) - Worked example B (half solved) - Exercise C (unsolved) - Connections to other cards - Common mistakes and misconceptions - Self-test questions - My notes and next steps - Sources and further reading - Answer key - Revision history - Standard and licence note ## When to use it Use this card whenever the outcome of your decision depends on the choices of other decision makers who are also thinking ahead: - Pricing and promotions against a small number of rivals (will a price cut start a price war?). - Capacity, entry and product launches where timing and commitment matter. - Supplier and partner relationships where monitoring is costly and trust is at stake. - Bidding in auctions and tenders, from procurement reverse auctions to acquisitions. - Negotiations, where each side's walk-away option shapes the deal (mba/mba-negotiation-guide). If there is only uncertainty about nature (weather, demand) and no strategic opponent, use a decision tree instead. ## The idea in plain language A game is any situation with **players**, **strategies** (complete plans of action) and **payoffs** that depend on what everyone chooses. Von Neumann and Morgenstern (1944) founded the field; Nash (1950, 1951) gave it the solution concept used everywhere today: a **Nash equilibrium** is a set of strategies, one per player, such that no player can do better by changing their own strategy while the others keep theirs. It is a point of mutual best responses, where no one has regrets. Three lessons recur across MBA cases. First, individually rational choices can produce a collectively bad outcome: in the prisoner's dilemma both firms cut prices and both earn less. Second, the shadow of the future changes behaviour: when firms meet again and again, cooperation can be sustained by the threat of future punishment (Friedman 1971; Axelrod 1984). Third, in sequential games what matters is not what a player threatens but what it would actually do when the moment comes; Schelling (1960) showed that limiting your own options, if done visibly and irreversibly, can change what your rival expects and therefore what it does. ## Formulas and notation **Nash equilibrium.** A strategy profile $s^{*} = (s_1^{*}, \dots, s_n^{*})$ is a Nash equilibrium if for every player $i$ and every alternative strategy $s_i$: $$u_i(s_i^{*}, s_{-i}^{*}) \ge u_i(s_i, s_{-i}^{*})$$ **Strictly dominant strategy.** $s_i^{d}$ is strictly dominant if $u_i(s_i^{d}, s_{-i}) > u_i(s_i, s_{-i})$ for every other $s_i$ and every $s_{-i}$. **Mixed-strategy equilibrium in a 2 x 2 game.** Each player mixes so that the other is indifferent. If the column player plays its first strategy with probability $q$, the row player is indifferent between its rows when $$q\,a_{11} + (1 - q)\,a_{12} = q\,a_{21} + (1 - q)\,a_{22}$$ and symmetrically the row player's probability $p$ makes the column player indifferent. **Repeated prisoner's dilemma with grim trigger.** With temptation $T$, reward for cooperation $R$, punishment $P$, sucker payoff $S$ ($T > R > P > S$) and discount factor $\delta$, cooperation is sustainable when $$\frac{R}{1 - \delta} \ge T + \frac{\delta P}{1 - \delta} \quad \Longleftrightarrow \quad \delta \ge \frac{T - R}{T - P}$$ **Discount factor from a rate and a continuation probability:** $$\delta = \frac{p_c}{1 + r}$$ **First-price sealed-bid auction**, $n$ bidders with independent private values uniform on $[0, \bar{v}]$, symmetric equilibrium bid and expected revenue (equal in the first- and second-price formats by revenue equivalence): $$b(v) = \frac{n - 1}{n}\,v, \qquad E[\text{revenue}] = \frac{n - 1}{n + 1}\,\bar{v}$$ | Symbol | Meaning | |---|---| | $i$, $-i$ | A player, and all players other than $i$ | | $s_i$, $s^{*}$ | A strategy of player $i$, an equilibrium strategy profile | | $u_i(\cdot)$ | Payoff (utility, profit) of player $i$ | | $a_{jk}$ | Row player's payoff in row $j$, column $k$ | | $p$, $q$ | Mixing probabilities of the row and column player | | $T$, $R$, $P$, $S$ | Temptation, reward, punishment and sucker payoffs in the prisoner's dilemma | | $\delta$ | Discount factor per period, between 0 and 1 | | $r$, $p_c$ | Discount rate per period, probability the game continues to the next period | | $n$, $v$, $\bar{v}$, $b(v)$ | Number of bidders, a bidder's private value, upper bound of values, equilibrium bid | ## Normal form, dominance and Nash equilibrium A **normal-form** (strategic-form) game is a payoff matrix: rows are one player's strategies, columns the other's, and each cell lists (row payoff, column payoff). It fits simultaneous moves, or moves where neither side sees the other's choice in time. The procedure, in order: 1. Eliminate strictly dominated strategies (no rational player uses a strategy that is worse whatever the others do). Repeat until nothing more can be removed (iterated dominance). 2. Among what remains, find every cell where each player is playing a best response to the other: those are the pure Nash equilibria. 3. If there is none, look for a mixed-strategy equilibrium (Nash 1951 proved that every finite game has at least one equilibrium, possibly in mixed strategies). 4. If there are several, ask which one the players would coordinate on (focal points, Schelling 1960), or whether one player can move first and select it. **Classic structures to recognise in cases.** | Structure | Pattern | Business reading | |---|---|---| | Prisoner's dilemma | Each has a dominant strategy; the equilibrium is worse for both than mutual cooperation | Price wars, advertising wars, overcapacity | | Coordination game | Two pure equilibria, both prefer to match | Technology standards, launch dates, interoperability | | Battle of the sexes | Two pure equilibria, players disagree on which is better | A standard war where each prefers its own format | | Chicken | Two asymmetric equilibria, disaster if both stand firm | Capacity races, brinkmanship in negotiations | | Inspection game | No pure equilibrium, mixed only | Audits, quality inspection, fraud control | ## Mixed strategies In some games any predictable choice is exploited. If Northfield always inspects incoming supplier lots, suppliers always comply and inspection is wasted; if Northfield never inspects, suppliers cut corners. The equilibrium is to inspect with a probability that leaves the supplier exactly indifferent between complying and cutting corners, while the supplier complies with a probability that leaves Northfield indifferent between inspecting and trusting. The counter-intuitive result: **your mixing probability is determined by your opponent's payoffs, not your own.** If the penalty on a supplier caught cutting corners rises, the equilibrium inspection rate falls (the supplier needs less monitoring to stay honest), while the supplier's compliance rate is unchanged. Worked example B shows this in numbers. ## Repeated games and cooperation When the same players meet repeatedly with no known last round, strategies can condition on history. Under **grim trigger** a firm cooperates until the rival defects once and then defects forever; under **tit-for-tat** it starts cooperating and then copies the rival's last move. Friedman (1971) showed that cooperative outcomes can be equilibria of infinitely repeated games if players are patient enough (the "folk theorem" family of results). Axelrod (1984) ran computer tournaments in which tit-for-tat, a strategy that is nice (never defects first), retaliatory, forgiving and clear, earned the highest total score. Three practical conditions make cooperation more stable: frequent interaction (each period is short, so $\delta$ is high), good visibility of rivals' actions (cheating is detected quickly), and a long expected horizon (a rival planning to exit has little to lose). Firms observe these conditions in markets with published prices and few players, which is also why competition authorities watch such markets closely. Explicit agreements to fix prices are illegal in most jurisdictions; the theory explains why tacit cooperation can emerge and why it can break down, not how to arrange it. ## Sequential games and credible commitment An **extensive-form** game is a tree: nodes are decisions, branches are actions, leaves carry payoffs. Solve it by **backward induction**: at each final decision, pick the best action for the player who moves, replace the node with that payoff, and work back to the start. The result is a **subgame perfect equilibrium** (Selten 1965, 1975): a Nash equilibrium that does not rely on threats a player would not carry out when the time comes. **Entry deterrence** *(Example)*. Corvane Robotics considers entering a regional market for warehouse-robot maintenance where an incumbent earns USD 100m (payoffs in USD m, present value). If Corvane stays out it earns 0 and the incumbent keeps 100. If Corvane enters, the incumbent can fight (price war: Corvane -10, incumbent 20) or accommodate (Corvane 30, incumbent 50). ```mermaid flowchart TD E{"Corvane: enter or stay out?"} -->|"Stay out"| O["Payoffs (0, 100)"] E -->|"Enter"| I{"Incumbent: fight or accommodate?"} I -->|"Fight"| F["Payoffs (-10, 20)"] I -->|"Accommodate"| A["Payoffs (30, 50)"] C{"Incumbent first: commit to capacity at cost 15?"} -->|"Yes"| E2{"Corvane: enter or stay out?"} C -->|"No"| E E2 -->|"Stay out"| O2["Payoffs (0, 85)"] E2 -->|"Enter"| I2{"Incumbent with sunk capacity"} I2 -->|"Fight"| F2["Payoffs (-10, 35)"] I2 -->|"Accommodate"| A2["Payoffs (30, 30)"] ``` Payoffs read (Corvane, incumbent). **Without commitment**, once Corvane has entered, the incumbent prefers to accommodate (50 beats 20). The threat to fight is not credible, Corvane anticipates this, enters and earns 30; the incumbent earns 50. **With commitment.** Suppose the incumbent can first install extra capacity at a sunk cost of 15. With that capacity in place, fighting is cheap (the capacity is already paid for and can flood the market) while accommodating leaves it idle: the incumbent's payoffs become 35 for fighting and 30 for accommodating. Now fighting is the incumbent's best response to entry, Corvane anticipates a price war (-10), stays out, and the incumbent earns $100 - 15 = 85$, much more than 50. The commitment works because it is **visible, costly and irreversible**: Schelling's (1960) insight that reducing your own flexibility can improve your outcome. A promise or press release that can be withdrawn at no cost would not change Corvane's calculation. ## Auctions basics | Format | Rule | Bidding in equilibrium (private values) | |---|---|---| | English (ascending) | Price rises until one bidder remains | Stay in until the price reaches your value | | Second-price sealed bid (Vickrey 1961) | Highest bid wins, pays the second-highest bid | Bid your true value: a weakly dominant strategy | | First-price sealed bid | Highest bid wins, pays its own bid | Shade the bid below value; with $n$ uniform bidders bid $(n-1)/n$ of value | | Dutch (descending) | Price falls until someone accepts | Strategically the same as first-price sealed bid | **Revenue equivalence** (Vickrey 1961; Myerson 1981; Riley and Samuelson 1981): with risk-neutral bidders, independent private values drawn from the same distribution, the highest-value bidder winning and a bidder with the lowest possible value expecting zero surplus, the standard formats give the seller the same expected revenue. With four bidders whose values are uniform between 0 and USD 100k, expected revenue is $100 \times 3/5 = 60$ (USD k) in both formats; a bidder with value 80 bids 60 in the first-price auction. More bidders raise revenue: 33.33 with two bidders, 77.78 with eight. **Winner's curse.** When the item has a common value that each bidder only estimates (an oil tract, a company being sold), the winner is the bidder with the most optimistic estimate and tends to overpay unless it shades its bid. In acquisitions, ask: "If I win, what does that tell me about what the other bidders knew?" ## Assumptions and limits - **Common knowledge of rationality.** Equilibrium analysis assumes each player is rational and knows the others are. Real managers have limited attention and different beliefs; behavioural game theory documents systematic deviations. - **Payoffs are known.** Matrices hide estimation error. Use ranges and check whether the equilibrium changes. - **Multiple equilibria.** Theory often says what could happen, not which outcome will. History, focal points and communication select among them. - **Payoffs include everything players care about.** Fairness, reputation and career concerns belong in the payoffs; if left out, predictions fail. - **Discounting in repeated games** depends on how long a period is and how likely the relationship is to continue; both are judgements. - **Auction results** rely on risk neutrality and independent private values; risk aversion, collusion among bidders and affiliated values change the ranking of formats. ## How managers use it 1. **Look forward, reason back.** Before a price cut, map the rival's best response and your response to that. 2. **Change the game, not only the move.** Brandenburger and Nalebuff (1996) frame strategy as changing players, added values, rules, tactics and scope; meet-the-competition clauses, loyalty programmes and capacity choices change payoffs. 3. **Make commitments credible.** Sink the cost, sign the contract, make it public. Otherwise rivals will ignore it. 4. **Design monitoring as a mixed strategy.** Random audits at a rate set by the other side's incentives are cheaper than full inspection and better than none. 5. **Protect cooperation in repeated relationships.** With suppliers and partners, reward reliability, respond to defection proportionately, and forgive quickly to avoid endless retaliation. 6. **Choose auction formats and bidding rules deliberately.** As a buyer running a procurement auction, more qualified bidders lower prices more than clever format design; as a bidder, shade for the winner's curse. ## Worked example A (fully solved) **Setting** *(Example)*. Northfield and Rival A (a generic label, not a real company) are the two large outdoor-gear chains in a region. Each quarter each chooses a High or Low price for its seasonal range. Quarterly operating profit in USD m: | Northfield \ Rival A | High | Low | |---|---|---| | High | 10, 10 | 4, 14 | | Low | 14, 4 | 6, 6 | **Step 1: dominance.** If Rival A plays High, Northfield earns 14 with Low and 10 with High. If Rival A plays Low, Northfield earns 6 with Low and 4 with High. Low is strictly dominant for Northfield, and by symmetry for Rival A. **Step 2: Nash equilibrium.** (Low, Low) with payoffs (6, 6) is the unique Nash equilibrium. Both would prefer (High, High) at (10, 10): a prisoner's dilemma with $T = 14$, $R = 10$, $P = 6$, $S = 4$. **Step 3: repeated play.** If the chains expect to meet every quarter indefinitely, grim trigger sustains (High, High) when $\delta \ge (T - R)/(T - P) = (14 - 10)/(14 - 6) = 0.5$. Excel: `=(14-10)/(14-6)` returns 0.5. **Step 4: is the condition met?** With a quarterly discount rate of 3 percent and certainty that both stay in the market, $\delta = 1/1.03 = 0.971$: well above 0.5. If there is only a 60 percent chance the relationship continues each quarter (Rival A is rumoured to be for sale), $\delta = 0.6/1.03 = 0.583$: still above 0.5, but close. With a 50 percent continuation probability and a 10 percent rate, $\delta = 0.5/1.10 = 0.455$: below 0.5, and the high-price outcome is no longer self-enforcing. Excel: `=0.6/(1+0.03)` returns 0.583. **Step 5: check the logic with present values at $\delta = 0.9$.** Cooperating forever is worth $10/(1 - 0.9) = 100$. Deviating once gives $14 + 0.9 \times 6/(1 - 0.9) = 14 + 54 = 68$. Cooperation wins. Excel: `=10/(1-0.9)` and `=14+0.9*6/(1-0.9)`. **Interpretation.** A one-shot analysis predicts a price war; repeated interaction can sustain higher prices as long as both chains expect to be around and can see each other's prices. The managerial risk signal is anything that lowers $\delta$: a rival under financial stress, a pending sale, or a shrinking market. The manager's job is not to coordinate with the rival (which competition law forbids) but to anticipate when price competition will intensify and to compete on differentiation, service and cost so that less depends on the price game. ## Worked example B (half solved) **Setting** *(Example)*. Northfield buys insulated jackets from a contract manufacturer. Each lot, Northfield chooses to Inspect (costly) or Trust; the supplier chooses to Comply with the specification or Cut corners. Payoffs per lot, USD k, (Northfield, supplier): | Northfield \ Supplier | Comply | Cut | |---|---|---| | Inspect | 8, 6 | 6, 2 | | Trust | 10, 6 | 0, 10 | 1. Check that there is no pure Nash equilibrium: from (Trust, Comply) the supplier switches to Cut; from (Trust, Cut) Northfield switches to [ ]; from (Inspect, Cut) the supplier switches to [ ]; from (Inspect, Comply) Northfield switches to Trust. 2. Supplier indifference: Comply pays 6 whatever Northfield does; Cut pays $2p + 10(1 - p)$ where $p$ is the probability Northfield inspects. Set equal: $6 = 10 - 8p$, so $p = $ [ ]. 3. Northfield indifference: Inspect pays $8q + 6(1 - q)$; Trust pays $10q$, where $q$ is the probability the supplier complies. Set equal: $6 + 2q = 10q$, so $q = $ [ ]. 4. Expected payoffs in equilibrium: Northfield [ ], supplier [ ]. 5. Suppose a contract penalty makes the supplier's payoff when caught cutting corners $-6$ instead of 2. The new inspection probability is $p = $ [ ] and the supplier's compliance probability [changes / does not change]. 6. In one sentence: why does a tougher penalty change Northfield's inspection rate rather than the supplier's compliance rate? The chart plots Northfield's expected payoff from each action against the supplier's compliance probability $q$. The lines cross where Northfield is indifferent. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Northfield expected payoff from Inspect and Trust as the supplier's compliance probability rises (fictional inspection game)", "width": 380, "height": 220, "data": {"values": [ {"q": 0.0, "Action": "Inspect", "Payoff": 6.0}, {"q": 0.1, "Action": "Inspect", "Payoff": 6.2}, {"q": 0.2, "Action": "Inspect", "Payoff": 6.4}, {"q": 0.3, "Action": "Inspect", "Payoff": 6.6}, {"q": 0.4, "Action": "Inspect", "Payoff": 6.8}, {"q": 0.5, "Action": "Inspect", "Payoff": 7.0}, {"q": 0.6, "Action": "Inspect", "Payoff": 7.2}, {"q": 0.7, "Action": "Inspect", "Payoff": 7.4}, {"q": 0.8, "Action": "Inspect", "Payoff": 7.6}, {"q": 0.9, "Action": "Inspect", "Payoff": 7.8}, {"q": 1.0, "Action": "Inspect", "Payoff": 8.0}, {"q": 0.0, "Action": "Trust", "Payoff": 0.0}, {"q": 0.1, "Action": "Trust", "Payoff": 1.0}, {"q": 0.2, "Action": "Trust", "Payoff": 2.0}, {"q": 0.3, "Action": "Trust", "Payoff": 3.0}, {"q": 0.4, "Action": "Trust", "Payoff": 4.0}, {"q": 0.5, "Action": "Trust", "Payoff": 5.0}, {"q": 0.6, "Action": "Trust", "Payoff": 6.0}, {"q": 0.7, "Action": "Trust", "Payoff": 7.0}, {"q": 0.8, "Action": "Trust", "Payoff": 8.0}, {"q": 0.9, "Action": "Trust", "Payoff": 9.0}, {"q": 1.0, "Action": "Trust", "Payoff": 10.0} ]}, "mark": {"type": "line", "point": true}, "encoding": { "x": {"field": "q", "type": "quantitative", "title": "Probability the supplier complies (q)"}, "y": {"field": "Payoff", "type": "quantitative", "title": "Northfield expected payoff (USD k per lot)"}, "color": {"field": "Action", "type": "nominal", "title": "Northfield action"} } } ``` ## Exercise C (unsolved) **Setting** *(Example)*. Two makers of technical shell fabric each choose a capacity of 25, 35 or 50 (thousand metres). Price is $100 - q_1 - q_2$ net of marginal cost, so each firm's profit is $(100 - q_1 - q_2)\,q_i$ (USD k). 1. Build the 3 x 3 payoff matrix. Excel, with the row capacity in A2 and the column capacity in B1: `=(100-$A2-B$1)*$A2` for the row firm. 2. Find every pure-strategy Nash equilibrium. Is any strategy strictly dominated? 3. Now let firm 1 choose its capacity first and visibly, with firm 2 responding. Solve by backward induction. Which capacity does firm 1 choose, and what are both payoffs? 4. Compare the payoffs if both chose 25. Which classic game structure does the simultaneous version resemble, and why? 5. Name one real action that would make firm 1's early capacity choice credible. ## Connections to other cards | Card or template | Connection | |---|---| | mba/mba-market-structure-card | Cournot and Bertrand are Nash equilibria; Stackelberg is a sequential game | | mba/mba-pricing-card | Price wars, matching guarantees and promotions as games | | mba/mba-negotiation-guide | Walk-away options, commitment and credibility at the bargaining table | | mba/mba-decision-trees-card | Trees against nature; backward induction is the same procedure | | mba/mba-five-forces-card | Rivalry intensity is the outcome of the pricing game | | mba/mba-competitive-advantage-card | Commitment and first-mover advantages as sources of advantage | | mba/mba-microeconomics-guide | Course roadmap and where this card sits | ## Common mistakes and misconceptions 1. **Reading the wrong payoff.** In each cell the first number belongs to the row player. Check before underlining. 2. **Treating Nash equilibrium as the best outcome.** It is a stable outcome, often worse for everyone than an alternative (the prisoner's dilemma). 3. **Setting your own mix from your own payoffs.** In a mixed equilibrium each player's probabilities make the other player indifferent. 4. **Believing every threat.** In sequential games, ignore threats the player would not carry out when the time comes. 5. **Assuming repetition always produces cooperation.** It needs patience (high $\delta$), visibility and no known end date; with a known last round, backward induction can unravel cooperation. 6. **Bidding your value in a first-price auction.** You win and earn nothing; bid shading is rational. 7. **Ignoring the winner's curse** in common-value auctions such as acquisitions. 8. **Confusing a dominant strategy with a Nash equilibrium strategy.** Every dominant-strategy equilibrium is a Nash equilibrium, but most Nash equilibria do not involve dominant strategies. 9. **Forgetting that commitment must be costly to be believed.** Cheap talk rarely moves a rational rival. 10. **Using game theory to plan collusion.** Agreements with competitors on price or output are illegal in most jurisdictions; use the analysis to anticipate rivals, not to coordinate with them. ## Self-test questions 1. Define a Nash equilibrium in one sentence. 2. In the matrix of Worked example A, why is (High, High) not an equilibrium even though both prefer it? 3. A prisoner's dilemma has $T = 20$, $R = 12$, $P = 5$, $S = 0$. What discount factor sustains cooperation under grim trigger? 4. The quarterly discount rate is 2 percent and the probability the relationship continues each quarter is 0.8. What is $\delta$, and is it enough for question 3? 5. Name two classic 2 x 2 games with two pure equilibria and give a business example of each. 6. In Worked example B, what happens to the supplier's compliance probability if Northfield's gain from trusting a compliant supplier rises from 10 to 12? 7. What makes a commitment credible? Give three properties. 8. Why is bidding your true value a weakly dominant strategy in a second-price sealed-bid auction? 9. With six bidders and values uniform on [0, 100], what is the equilibrium first-price bid of a bidder with value 90, and the seller's expected revenue? 10. What is the winner's curse, and in which kind of auction does it arise? ## My notes and next steps | Item | My note | |---|---| | Situation I modelled and its classic structure | [Prisoner's dilemma / coordination / chicken / inspection] | | Players, strategies and source of payoffs | [One line] | | Equilibrium and whether it is credible | [One line] | | One way to change the game in my favour | [One line] | | Self-test score and misses logged | [x / 10, logged in mba/mba-error-log on YYYY-MM-DD] | | Next card | [mba/mba-market-structure-card / mba/mba-negotiation-guide] | ## Sources and further reading Primary works: - Axelrod, R. (1984). *The Evolution of Cooperation*. Basic Books. - Brandenburger, A. M., and Nalebuff, B. J. (1996). *Co-opetition*. Currency Doubleday. - Friedman, J. W. (1971). A non-cooperative equilibrium for supergames. *The Review of Economic Studies*, 38(1), 1-12. - Myerson, R. B. (1981). Optimal auction design. *Mathematics of Operations Research*, 6(1), 58-73. - Nash, J. F. (1950). Equilibrium points in n-person games. *Proceedings of the National Academy of Sciences*, 36(1), 48-49. - Nash, J. F. (1951). Non-cooperative games. *Annals of Mathematics*, 54(2), 286-295. - Riley, J. G., and Samuelson, W. F. (1981). Optimal auctions. *The American Economic Review*, 71(3), 381-392. - Schelling, T. C. (1960). *The Strategy of Conflict*. Harvard University Press. - Selten, R. (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. *Zeitschrift für die gesamte Staatswissenschaft*, 121, 301-324 and 667-689. - Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in extensive games. *International Journal of Game Theory*, 4(1), 25-55. - Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. *The Journal of Finance*, 16(1), 8-37. - von Neumann, J., and Morgenstern, O. (1944). *Theory of Games and Economic Behavior*. Princeton University Press. Textbooks (further reading, any recent edition): Dixit, Skeath and McAdams, *Games of Strategy*; Dixit and Nalebuff, *The Art of Strategy*; Gibbons, *Game Theory for Applied Economists*; Besanko, Dranove, Shanley and Schaefer, *Economics of Strategy*. Open resource: OpenStax, *Principles of Economics* (3rd edition, CC BY-NC-SA 4.0), oligopoly chapter (prisoner's dilemma), openstax.org. ## Answer key **Worked example B.** (1) From (Trust, Cut) Northfield switches to Inspect (6 beats 0); from (Inspect, Cut) the supplier switches to Comply (6 beats 2). (2) $p = 4/8 = 0.5$. (3) $q = 6/8 = 0.75$. (4) Northfield $8 \times 0.75 + 6 \times 0.25 = 7.5$ (equal to $10 \times 0.75$); supplier 6. (5) $6 = -6p + 10(1 - p)$ gives $p = 4/16 = 0.25$; the compliance probability does not change, because it is set by Northfield's payoffs, which did not change. (6) Each player's mix must keep the other indifferent; the penalty enters the supplier's payoffs, so it moves the probability Northfield needs to keep the supplier indifferent. **Exercise C.** (1) Payoffs (row, column): | Row \ Column | 25 | 35 | 50 | |---|---|---|---| | 25 | 1250, 1250 | 1000, 1400 | 625, 1250 | | 35 | 1400, 1000 | 1050, 1050 | 525, 750 | | 50 | 1250, 625 | 750, 525 | 0, 0 | (2) The unique pure Nash equilibrium is (35, 35) with 1,050 each. Yes: 50 is strictly dominated by 35 (1,400 beats 1,250, 1,050 beats 750 and 525 beats 0). Once 50 is removed, 25 is strictly dominated by 35 (1,400 beats 1,250 and 1,050 beats 1,000), so iterated dominance alone leads to (35, 35). Best replies: 35 to 25 and to 35, 25 to 50. (3) If firm 1 picks 25, firm 2 replies 35 (1,000 for firm 1); if 35, firm 2 replies 35 (1,050); if 50, firm 2 replies 25 (1,250). Firm 1 commits to 50 and earns 1,250; firm 2 earns 625, which mirrors the Stackelberg result in mba/mba-market-structure-card. Note that the leader gains by committing to 50, a capacity that is strictly dominated in the simultaneous game: commitment changes what the rival believes, which is its whole value. (4) Both choosing 25 gives 1,250 each, better for both than the equilibrium 1,050: a prisoner's-dilemma structure, because each firm's best reply to 25 is to expand. (5) Ordering and paying for the equipment, signing long-term supply contracts for the output, or building the plant before the rival decides. **Self-test.** 1. A set of strategies, one per player, in which no player can gain by changing only its own strategy. 2. Each firm's best response to High is Low (14 beats 10), so each would deviate. 3. $\delta \ge (20 - 12)/(20 - 5) = 8/15 = 0.533$. 4. $\delta = 0.8/1.02 = 0.784$, above 0.533, so cooperation is sustainable. 5. Coordination (two firms adopting the same connector standard) and battle of the sexes (each prefers its own standard but both prefer a common one to none); chicken (a capacity race) also has two pure equilibria. 6. Northfield's indifference becomes $8q + 6(1 - q) = 12q$, so $q = 6/10 = 0.6$: the supplier needs to comply less often to keep Northfield indifferent. Northfield's inspection rate is unchanged. 7. Visible (the rival can observe it), costly or irreversible (it cannot be undone cheaply), and it changes the committing player's own payoffs so that carrying out the threat becomes its best response. 8. Your bid only decides whether you win, not what you pay. Bidding above value risks winning at a price above value; bidding below risks losing an item you would have bought at a profitable price. Bidding your value never does worse. 9. Bid $90 \times 5/6 = 75$; expected revenue $100 \times 5/7 = 71.43$. 10. In a common-value auction, the winner is the bidder with the most optimistic estimate and so tends to overpay unless it shades its bid for that selection effect. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First pass through the card | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Modelled the price war in my strategy case as a repeated game | ## Standard and licence note Based on the standard treatment of non-cooperative game theory in MBA microeconomics and strategy courses, drawing on the primary works listed above (von Neumann and Morgenstern 1944, Nash 1950 and 1951, Schelling 1960, Vickrey 1961, Selten 1965 and 1975, Friedman 1971, Myerson 1981, Axelrod 1984). The text, the payoff matrices, the worked examples and the questions are original; no textbook text or published case is copied or paraphrased, and the fictional companies are not real businesses. The template is independent and not affiliated with or endorsed by any business school. Template licence: MIT.