Game Theory Foundations

Competitors move because they expect you to move

ClearPeak's rooftop solar rebate program was meant to slow customer defections. Within six weeks, two neighboring utilities matched the rebate and a national installer bundled batteries at zero upfront cost. Dr. Elena Vasquez realized ClearPeak was not setting price in a vacuum; it was playing a strategic game where rivals respond to beliefs about ClearPeak's next move. Game theory (the study of interdependent decisions when outcomes depend on what others choose) gives vocabulary for that interdependence: players, strategies, payoffs, and equilibrium concepts.

Regulated utilities still face strategic rivals: distributed solar installers, demand-response aggregators, and neighboring utilities competing for commercial load. The State Public Utilities Commission (PUC, the regulator that approves rates and resource plans) watches whether ClearPeak's moves are pro-competitive or predatory. Game theory does not replace regulation; it clarifies when cooperative outcomes are stable and when price wars or capacity races destroy value.

ClearPeak Energy is a regulated regional electric utility serving 1.2 million residential and commercial customers across twelve counties and the anchor organization for ECO 101. The utility faces retiring 2,400 MW of coal while adding 1,800 MW of utility-scale solar and battery storage by 2030, peak summer demand near 8,500 MW, and an average residential bundled rate of $0.118/kWh (kilowatt-hour, enough electricity to run ten 100-watt bulbs for one hour). Chief Economist Dr. Elena Vasquez, Regulatory Affairs VP Tom Bradley, and Grid Planning Director Amara Okafor use microeconomic tools for rate design, capacity planning, competitive response, and State Public Utilities Commission (PUC) testimony. Marginal generation costs differ sharply: legacy coal near $0.042/kWh, new solar near $0.031/kWh, and gas peakers near $0.067/kWh when scarcity bites.

Every lesson applies supply, demand, elasticity, marginal analysis, market structure, or incentive design to decisions ClearPeak leaders actually face: when to retire plants, how to price time-of-use tiers, how to bid in capacity auctions, and how to respond when rooftop solar erodes sales.

This lesson covers normal-form games, dominant strategies, Nash equilibrium, and sequential moves. Later lessons apply the same logic to pricing wars, auctions, and hidden information.

Players, strategies, and payoffs

A player is a decision-maker (ClearPeak, SolarPeak installer coalition, commercial customer). A strategy is a complete plan of action (match rebate, undercut by $200/kW, hold price). Payoffs are profits, market share, or regulatory approval scores. Represent simple simultaneous games in a payoff matrix with rows and columns for strategies.

Dominant strategy and dominated strategies

A strategy is dominant if it yields the best payoff regardless of rival action. If no dominant strategy exists, players must reason about rivals' choices. Iterated elimination of dominated strategies removes obviously bad options before solving for equilibrium.

Numbers are illustrative share points. Rebate helps if rival ignores; if rival matches, both spend margin with limited share gain.

Nash equilibrium

A Nash equilibrium (named for John Nash) is a strategy profile where no player can improve payoff by unilaterally deviating. In the rebate matrix above, (Offer, Match) and (Hold, Ignore) can both be Nash equilibria depending on payoffs. Multiple equilibria mean history and expectations matter.

Sequential games and credible threats

In Stackelberg (leader-follower) models, ClearPeak moves first on time-of-use rate design; rivals respond with solar lease pricing. A threat to cut rates below marginal cost may not be credible if regulators cap losses. Draw game trees: nodes, branches, backward induction to find subgame-perfect equilibrium.

Cooperation, repetition, and regulation

One-shot prisoners' dilemma pushes rivals toward aggressive pricing. Repeated games allow tacit cooperation if future punishment is valued. Utilities rarely collude illegally; instead they signal through public filings. PUC oversight changes payoffs: predatory pricing invites investigation.

Simultaneous choice: ClearPeak offers $800/kW rebate or holds; installer coalition matches or focuses on battery upsell.

Part A: Payoff matrix

| | Installer match | Installer battery focus | | ClearPeak rebate | (1.2% share gain each, -$18M cost) | (ClearPeak +2.1% share, -$18M) | | ClearPeak hold | (Installer +1.5% share) | (Status quo margins) |

Part B: Nash read

If installer always matches, ClearPeak's rebate spends $18M for ~1.2% share (~14,400 customers). At $14/month contribution, annual gross ≈ $2.4M vs $18M spend year one. Rebate is dominated unless retention of high-margin commercial load included.

Part C: Sequential alternative

ClearPeak announces time-of-use (TOU, higher price on peak hours) first; installer prices solar+storage against peak savings. Backward induction: installer targets peak kWh savings; ClearPeak designs TOU to retain margin without pure rebate race.

Part D: Managerial read

Tom Bradley presents game tree to PUC: cooperative technology-neutral efficiency programs beat dueling rebates.

GridCo accused a rival of withholding peakers in a heat wave. Game theory clarifies: unilateral withholding is profitable only if others do not undercut. With multiple entrants, Nash logic often favors dispatch, not conspiracy. ClearPeak documents peaker availability hourly for regulatory defense.

In a 2×2 game, ClearPeak's strategy A yields 4 if rival plays X, 1 if Y. Strategy B yields 3 regardless. Dominant strategy?

Solution

B is dominant: 3 > 1 when rival plays Y, and compare A vs B when rival plays X (need A payoff vs 3). If A gives 4 vs X, no dominant strategy; if A gives 2 vs X, B dominates. State your assumed payoffs. Check: define dominant as best against each rival move ✓

Why might ClearPeak prefer sequential TOU announcement to simultaneous rebate matching?

Solution

Leader sets framing for peak savings competition; avoids burned cash in mutual rebate Nash where both spend and share stabilizes. Regulators see price structure reform, not subsidy war.

Key takeaways

• Strategic rivals respond to ClearPeak moves; model interdependence with game theory.
• Nash equilibrium identifies stable strategy pairs with no profitable unilateral deviation.
• Dominant strategies simplify decisions; many utility games require full equilibrium analysis.
• Sequential games and credible threats matter for TOU and resource planning.
• Regulatory payoffs belong in the matrix, not only market share.

1. Sketch a payoff matrix for ClearPeak demand-response partnership vs go-alone.
2. Identify one non-credible threat in a proposed price war memo.
3. Continue to Lesson 2: Pricing Games and Competitive Response.

Applying Game Theory Foundations at ClearPeak scale

When ClearPeak Energy evaluates game theory foundations, Dr. Elena Vasquez starts from operational facts: 1,200,000 customers, peak demand near 8,500 MW, residential bundled rate $0.118/kWh, and a portfolio transition that retires 2,400 MW of coal while adding 1,800 MW of solar. game theory, auctions, and information economics is not textbook decoration; it is how Tom Bradley prepares State Public Utilities Commission (PUC) filings and how Amara Okafor ranks transmission and storage options under binding capital budgets.

Graph (described in prose): Game Theory Foundations at ClearPeak. Imagine a standard microeconomics diagram with quantity (megawatt-hours or customer count, depending on the decision) on the horizontal axis and price ($/kWh) or marginal cost ($/kWh) on the vertical axis. The demand curve slopes downward: at higher retail rates, customers conserve, shift load to off-peak hours, or install rooftop solar. The supply curve in the short run reflects rising marginal cost as ClearPeak dispatches coal, combined-cycle gas, and expensive peakers. Equilibrium is where quantity demanded equals quantity supplied at a price regulators allow; in regulated markets, equilibrium is a negotiated outcome, not only a frictionless auction. When ${title.toLowerCase()} changes, curves shift: new solar lowers long-run supply cost; heat waves shift demand right; competitor solar leases shift demand left for utility energy. Shaded consumer surplus and producer surplus (or deadweight loss when prices depart from marginal cost) translate directly into affordability testimony and earnings impacts.

Work a magnitude check. Suppose a policy tied to game theory foundations moves residential sales by 1% at current scale. One percent of 1,200,000 customers is 12,000 accounts. At roughly 900 kWh per month average use and $0.118/kWh, a 1% quantity change moves monthly revenue by about $1.3 million before fuel cost adjustments. Executives who skip arithmetic like this debate symbols without stakes.

Extended ClearPeak scenario: regulatory and competitive read

Imagine ClearPeak's quarterly review on game theory foundations. Finance asks whether a rate increase recovers rising gas peaker costs. Operations asks whether demand response can defer a $400 million substation upgrade. Commercial customers ask for advanced metering discounts. Rooftop solar installers tell regulators ClearPeak exercises market power. A weak game theory, auctions, and information economics answer addresses only one audience. A strong answer links curves, elasticities, and marginal costs to each stakeholder's metric.

Dr. Vasquez uses a three-panel narrative. Panel one: short-run dispatch when peak load hits 8,500 MW and peakers set marginal cost near $0.067/kWh. Panel two: long-run portfolio when solar at $0.031/kWh displaces coal at $0.042/kWh plus carbon compliance. Panel three: competitive fringe where distributed solar at $0.09/kWh effective price steals high-margin afternoon sales. Game Theory Foundations supplies vocabulary to keep the panels consistent.

Numerical discipline example: if price elasticity of residential demand is -0.35 (a 1% price rise cuts quantity about 0.35%), a 4% rate increase reduces energy sales roughly 1.4% in the short run. Combined with weather normalization, Elena reports a bounded revenue forecast instead of pretending demand is fixed. Regulators punish utilities that ignore elasticity in revenue requirement testimony.

Technical mechanics and reconciliation checks

For game theory foundations, ClearPeak analysts show work the way accountants show trial balances. A supply table lists plant, capacity MW, heat rate, variable O&M, fuel cost, and marginal cost per MWh (megawatt-hour). A demand table lists customer class, price, quantity, and expenditure. Equilibrium checks that quantity demanded equals scheduled dispatch within reserve margin rules. Elasticity checks recompute percent changes with the same denominator conventions used in the tariff filing.

Use explicit formula lines before plugging numbers. Elasticity = percent change in quantity demanded divided by percent change in price. Marginal cost = change in total cost divided by change in output. Marginal revenue = change in total revenue divided by change in quantity sold. Consumer surplus approximates the area below demand and above price for the units consumed. When lessons use linear demand shortcuts, state the assumption: "linear between two observed tariff points."

Spreadsheet grain matters. Utility models often run hourly for dispatch, monthly for billing, and annual for regulatory revenue requirements. Game Theory Foundations fails silently when rows mix grains. Elena requires a grain column in every workbook: hour, month, customer-month, or plant-year.

Common executive questions (and disciplined answers)

Executives ask short questions that need long disciplined answers. "Can we pass fuel costs through?" maps to allowed riders, elasticity, and affordability indices, not anger on social media. "Will solar kill the utility?" maps to cross-price elasticity with distributed energy and fixed cost recovery. "Why not cut rates to grow?" maps to marginal revenue sign when |elasticity| < 1. "What is fair return?" maps to allowed revenue requirement and cost of capital, not last year's earnings plus 10%.

ClearPeak's credible answer format for game theory foundations is three bullets: recommendation, key elasticities or marginal costs behind it, and what evidence would reverse the view within two quarters. A fourth bullet names deadweight loss or equity tradeoffs when policy moves price away from marginal cost.

Practice the translation loop until habit: business question → curves and elasticities → quantity and revenue arithmetic → stakeholder table → filing language. Broken loops produce pretty charts that fail cross-examination.

Practice extension: graph and arithmetic self-check

Before re-reading solutions, sketch four items on paper. Item one: draw (in words) demand and supply for ClearPeak summer peak hours with labels. Item two: write one shift that increases price and one that decreases quantity without a price change. Item three: compute percent ΔQ and percent ΔP for a scenario in the lesson and verify elasticity sign. Item four: state who gains and who loses in surplus terms.

Compare your sketch to the worked example. Gaps tell you what to re-read. If you work outside utilities, substitute your product but keep the same structure: define market, state margins, show equilibrium, stress-test with elasticity.

Connection to ACC 101, MKT 202, and capstone design

ACC 101 taught you to reconcile statements; ECO 101 teaches you to reconcile marginal stories with average costs regulators allow. MKT 202 taught evidence ladders; here the ladder is descriptive load research → elasticity estimation → pricing experiment or pilot tariff → regulatory approval. Unit six capstone on designing incentives expects you to combine game theory, auctions, and information economics with game theory and externality tools from earlier units.

Integrated narrative example: ClearPeak proposes a peak-pricing pilot (MKT-style segmentation), estimates elasticity −0.35 (ECO 101 Unit 2), models revenue with marginal cost dispatch (Unit 3), and defends fairness to the PUC (Unit 6). Courses compound when vocabulary and numbers stay consistent.

Deep dive: ClearPeak data definitions reused every month

Residential bundled rate includes energy, distribution, and mandated riders; pilots may unbundle for time-of-use. Peak demand is the highest hourly load in a month; coincident peak may determine transmission charges. Marginal cost of service for pricing studies uses forward-looking dispatch, not historical average embedded cost. Lost revenue from energy efficiency or solar is offset by decoupling mechanisms in some filings. Elasticity estimates separate weather, price, income, and appliance stock effects.

Definition drift fakes wins. If operations reports peak MW using one weather adjustment and finance uses another, game theory foundations recommendations flip. Elena publishes a one-page data dictionary before each major filing.

Monthly reconciliation: billed energy ≈ generation net losses ± inventory; revenue ≈ Σ quantity × tariff by class; marginal cost tables sum to dispatch cost within rounding. Elasticity replays on holdout months. When reconciliations fail, fix data before arguing policy.

Managerial judgment prompts for Game Theory Foundations

1. If elasticity is inelastic short run but elastic long run, how should ClearPeak sequence a multi-year rate path?
2. If marginal solar cost is below coal but fixed grid costs rise, is average cost or marginal cost the right public narrative?
3. Which stakeholder loses most if ClearPeak underestimates cross-price elasticity with rooftop solar?
4. What observable would convince you the demand curve shifted versus movement along the curve?
5. When does surplus language help regulators and when does it sound like economist jargon?

Write ninety-word memo answers using ClearPeak numbers. This converts lesson prose into testimony reflexes.

Additional study path: compare this lesson's practice problem to the worked example. Identify one assumption that changed elasticity or marginal cost and explain how the decision flips. Capstone integration is intentional; reuse ClearPeak names and units across units.

Numerical walk-through: peak hour dispatch

Consider a summer peak hour with 8,500 MW demand. ClearPeak dispatches 3,200 MW coal at $0.042/kWh variable, 3,800 MW combined-cycle gas at $0.055/kWh, 800 MW solar at near-zero variable cost, and 700 MW peakers at $0.067/kWh. The marginal unit sets price in competitive benchmarks; in regulation, the filing may use average revenue requirement. Weighted average variable cost ≈ (3200×0.042 + 3800×0.055 + 800×0.005 + 700×0.067) / 8500 ≈ $0.046/kWh before T&D (transmission and distribution).

If game theory foundations motivates shifting 200 MW from peak to off-peak via time-of-use pricing, peaker runs drop, variable cost falls roughly 200×$0.067 = $13,400 per hour, plus avoided capacity charges if sustained. Demand response programs trade customer incentives against this savings. Elena documents both gross savings and participation costs; net benefit drives the filing.

Check: 3200+3800+800+700 = 8500 MW ✓. Any lesson using partial portfolios should show similar capacity checks.

Surplus, equity, and policy tradeoffs

Microeconomics is not only efficiency. Game Theory Foundations at ClearPeak intersects affordability programs for low-income households, equity when time-of-use shifts burden evening home use, and environmental justice when retired coal plants sit in vulnerable communities. Consumer surplus gains for average bills may hide losses for heat-vulnerable customers.

When lessons recommend raising price toward marginal cost, pair the recommendation with a transfer or assistance mechanism or explain why the PUC weights equity constraints. Dr. Vasquez tables deadweight loss of under-pricing peak energy alongside hardship metrics. Regulators accept tradeoffs stated clearly; they reject efficiency claims that ignore distributional facts.

For game theory, auctions, and information economics, practice writing one paragraph that a non-economist commissioner could read aloud. Avoid surplus jargon without translation: "customers who value afternoon cooling less than the cost of peaker plants would consume less under peak pricing, freeing capacity for hospitals and industrial employers."

Historical filing pattern (synthetic but consistent)

ClearPeak's 2024 time-of-use pilot covered 42,000 households. Control group average peak kWh fell 2.1% from weather normalization; pilot group fell 6.8%. Difference-in-differences estimate 4.7% peak reduction. With pilot peak price +18% versus control flat rate, arc elasticity ≈ 4.7/18 ≈ 0.26 in absolute value on the pilot margin (illustrative, not a policy filing). Revenue net of lost sales rose 1.2% because peak price uplift exceeded quantity loss on inelastic inframarginal hours.

Tom Bradley's lesson for game theory foundations: pilot evidence beats theory slides, but pilots need control groups and pre-registered metrics. Amara links observed peak reduction to deferred substation timing: 4.7% on 420 MW local peak ≈ 20 MW relief, extending asset life two years under stated loading rules.

Cross-price and income effects reminder

Game Theory Foundations rarely operates in isolation. Income elasticity matters when recession hits commercial load. Cross-price elasticity with rooftop solar matters when federal tax credits change. Cross-price elasticity with natural gas matters for dual-fuel customers. Elena keeps a small table of estimated elasticities by class: residential -0.35, commercial -0.55, industrial −0.22 short run.

When interpreting ClearPeak results, ask which elasticity dimension the decision uses. Price-only stories mislead if income or substitute prices moved simultaneously. Multiple-regression control variables belong in advanced courses; the managerial habit here is to name confounds even if you cannot quantify them yet.

Closing integration: from lesson to testimony bullet

Translate game theory foundations into a single testimony bullet ClearPeak could use: claim, mechanism, magnitude, caveat. Example structure: "We recommend expanding time-of-use because peak demand elasticity is modest short run but pilot evidence shows 4–7% peak kWh reduction at +18% peak price, deferring $40M substation spend if sustained two years, with low-income bill protection via tiered credits." Compare your bullet to the lesson takeaways. If magnitude or caveat is missing, deepen the quantitative thread before moving on.

Step-by-step elasticity replay (when relevant to Game Theory Foundations)

Suppose ClearPeak raises the residential energy charge from $0.095/kWh to $0.099/kWh, a 4.2% increase. Prior monthly sales averaged 720 GWh (gigawatt-hours). Estimated short-run own-price elasticity is -0.35. Expected quantity change ≈ -0.35 × 4.2% ≈ −1.47%. New sales ≈ 720 × (1 − 0.0147) ≈ 709.4 GWh.

Revenue before ≈ 720,000,000 kWh × $0.095 ≈ $68.4M per month (energy portion only). Revenue after ≈ 709,400,000 × $0.099 ≈ $70.2M. Despite lower volume, revenue rises because demand is inelastic (|ε| < 1). Tom Bradley uses this arithmetic in filings; Elena notes long-run elasticity may exceed −0.6, reversing the revenue gain over three years. Game Theory Foundations lessons should always pair short-run and long-run elasticity stories when pricing is involved.

Check: percent change formula uses consistent base (midpoint or initial); document which you use ✓

Marginal versus average cost at ClearPeak (cost ladder)

Average cost spreads fixed capital and O&M across all units; regulators use it in revenue requirements. Marginal cost tells Elena which plant runs next and what the last megawatt costs on a hot afternoon. Game Theory Foundations decisions fail when teams argue average while the grid dispatches marginal. For peak pricing pilots, marginal peaker cost near $0.067/kWh is the opportunity cost of an extra peak kWh.

Weighted check for variable dispatch stack (8500 MW example): coal+gas+solar+peaker shares sum to 100% ✓

Capstone linkage note (Game Theory Foundations in the full ECO 101 arc)

Unit one gave you curves; unit two gave elasticities; unit three gave costs and scale; unit four gave market power; unit five gave games and information; unit six gives policy design. Game Theory Foundations sits in that arc at ClearPeak: every formula should connect to a filing paragraph Tom Bradley could defend. When you draft recommendations, cite at least two prior-unit tools by name (for example, elasticity from Unit 2 plus externality pricing from Unit 6).

Dr. Vasquez's integrative standard: one page, five bullets, each bullet ties a concept to a number and a stakeholder. No bullet without magnitude. No magnitude without assumption. This is the difference between MBA fluency and undergraduate definition recall.