Chapter §1 · Bimatrix & Fixed Points
von Neumann (1928) · Nash (1950)
Normal-Form Games, Best Responses & Indifference
“Every finite game has an equilibrium point in mixed strategies, where each player's strategy is optimal against those of the others.” — John F. Nash Jr., PNAS (1950)
Theorem 1.1 (Mixed-Strategy Indifference Principle)
[Nash 1950; Luce & Raiffa §4.6]
In a 2 × 2 bimatrix game where Row plays Top (T) with probability p ∈ [0, 1] and Column plays Left (L) with probability q ∈ [0, 1], any interior Nash equilibrium (p*, q*) ∈ (0, 1)² requires each player to choose mixing weights that make the opponent strictly indifferent between pure actions:
E[u₁(T, q*)] = E[u₁(B, q*)] and E[u₂(p*, L)] = E[u₂(p*, R)] ∎
Equation (1.1) — Live Substituted Indifference & QRE
Executable Formula
q* = (a₂₂ − a₁₂) / [(a₁₁ − a₂₁) + (a₂₂ − a₁₂)], p* = (b₂₂ − b₂₁) / [(b₁₁ − b₁₂) + (b₂₂ − b₂₁)]
The Counterfactual Twist · Logit Quantal Response (McKelvey & Palfrey 1995)
| Row (I) \ Col (II) | Left (L) · prob q | Right (R) · prob 1−q |
|---|---|---|
| Top (T) · prob p |
u₁
u₂
|
u₁
u₂
|
| Bottom (B) · prob 1−p |
u₁
u₂
|
u₁
u₂
|
Pure Nash Equilibria2 Pure NE
Mixed Nash (p*, q*)(0.60, 0.40)
Logit QRE (p_λ, q_λ)(0.58, 0.42)
Expected Payoff (u₁, u₂)(1.20, 1.20)
Fig 1.1 — Left: Unit Square (p, q) Best-Response Step Functions, QRE Curves & Brown (1951) Fictitious Play Spiral · Right: Row Expected Utility E[u₁(q)]
Carmine: BR₁(q) · Prussian: BR₂(p) · Brass ◆: Nash / QRE
Sit in Player I's Chair — Play Against the Textbook
Rounds: 0 · Your Empirical p̂: — · Book Strategy: Fictitious Play Learner
Choose Top or Bottom to test whether Fictitious Play exploits your pattern.
Chapter §2 · Repeated Games & Shadow of the Future
Flood & Dresher (1950) · Axelrod (1984)
Iterated Prisoner's Dilemma & Folk Theorem Thresholds
“For cooperation to prove stable, the future must have a sufficiently large shadow.” — Robert Axelrod, The Evolution of Cooperation (1984)
Theorem 2.1 (Folk Theorem & Trigger Sustainability)
[Fudenberg & Maskin 1986; Axelrod 1984]
In an infinitely repeated Prisoner's Dilemma with stage payoffs T > R > P > S and continuation probability δ ∈ (0, 1), mutual cooperation (C, C) is sustainable as a Subgame Perfect Nash Equilibrium under Grim Trigger iff δ ≥ (T − R)/(T − P), and under Tit-for-Tat iff δ ≥ max((T − R)/(T − P), (T − R)/(R − S)).
V(C,C) = R / (1 − δ) ≥ T + δ P / (1 − δ) ∎
Equation (2.1) — Shadow of the Future & Trembling-Hand Noise
Executable Formula
δ*_Grim = (T − R) / (T − P), δ*_TFT = max((T − R)/(T − P), (T − R)/(R − S))
The Counterfactual Twist · Trembling-Hand Noise ε (Selten 1975 / Nowak & Sigmund 1993)
Duel Mean (u₁, u₂)(3.00, 3.00)
Cooperation Rates100% · 100%
Folk Threshold δ*_TFT0.67 (Sustainable)
Tournament ChampionGenerous TFT
Fig 2.1 — Top: Round-by-Round Punchcard Tape (● Cooperate, × Defect, ⚡ Noise Slip) · Bottom: Cumulative Payoff Trajectories
Seed #42
| Rank | Automaton Strategy | Mean Payoff / Rd | Coop Rate | Head-to-Head Wins | Behavioral Diagnosis under Noise ε |
|---|
Sit in Player I's Chair — Live Repeated Duel vs. Automaton II
Round 0 · Your Total: 0 · Opponent Total: 0
Play C or D against the selected Column Automaton.
Chapter §3 · Population Dynamics on Δ²
Maynard Smith (1973) · Taylor & Jonker (1978)
Evolutionarily Stable Strategies & the Replicator Simplex
“Natural selection does not require conscious foresight; strategy shares grow in proportion to their excess fitness above the population average.” — Taylor & Jonker, Math. Biosciences (1978)
Theorem 3.1 (Replicator–Mutator ODE & ESS Stability)
[Maynard Smith & Price 1973; Hofbauer & Sigmund 1998]
Let x = (x₁, x₂, x₃)ᵀ ∈ Δ² denote population shares across three phenotypes with payoff matrix A. Under proportional selection and uniform mutation μ ≥ 0, every interior Evolutionarily Stable Strategy (ESS) x* satisfying (x* − x)ᵀ A x > 0 for all x ≠ x* is a globally asymptotically stable rest point of the replicator dynamics.
dxᵢ/dt = xᵢ [ (Ax)ᵢ − xᵀAx ] + μ (1/3 − xᵢ) ∎
Equation (3.1) — Live Fitness Vector f = Ax & Velocity at x(0)
Executable Formula
fᵢ(x) = ∑ⱼ Aᵢⱼ xⱼ, f̄(x) = xᵀ A x, H(x) = −∑ᵢ xᵢ ln(xᵢ)
The Counterfactual Twist · Cyclic Non-Convergence & Mutation Floor μ
Initial State x(0)(0.55, 0.30, 0.15)
Terminal State x(T)(0.33, 0.33, 0.33)
Mean Fitness f̄(0) → f̄(T)0.04 → 0.08
Dynamical RegimeAttracting ESS
Fig 3.1 — Left: Barycentric Simplex Δ² Vector Field & RK4 Trajectory (Click anywhere in triangle to launch new x(0)) · Right: Share Time-Series xᵢ(t)
○ Start x(0) · ◆ Terminal x(T)
Chapter §4 · Sequential Trees & Subgame Perfection
Zermelo (1913) · Kuhn (1953) · Selten (1965)
Extensive-Form Trees, Credible Threats & Backward Induction
“A Nash equilibrium is subgame perfect if and only if the players' strategies induce an equilibrium in every subgame — ruling out non-credible threats.” — Reinhard Selten (1965)
Theorem 4.1 (Zermelo–Kuhn Backward Induction & Fehr–Schmidt Utility)
[Selten 1965; Fehr & Schmidt 1999]
Every finite perfect-information extensive game has a pure-strategy Subgame Perfect Nash Equilibrium (SPNE) found by folding terminal payoffs backward from the deepest decision nodes. When players exhibit Fehr–Schmidt inequity aversion (α envy toward disadvantageous splits, β guilt toward advantageous splits), utilities transform as:
Uᵢ(x₁, x₂) = xᵢ − α max(xⱼ − xᵢ, 0) − β max(xᵢ − xⱼ, 0) ∎
Equation (4.1) — Live Backward-Induction Fold & Utility Shift
Executable Formula
V(h) = arg max_{a ∈ A(h)} U_{P(h)}(BI(h, a))
The Counterfactual Twist · Why Humans Reject Stingy Ultimatums & Pass in Centipede
Subgame Perfect PathGreedy → Accept
Material Outcome (x₁, x₂)(8.0, 2.0)
Social Utility (U₁, U₂)(8.00, 2.00)
Equilibrium Shift vs. α=0Classical SPNE Holds
Fig 4.1 — Extensive Game Tree Γ · Bold Carmine/Prussian Edges: Active Subgame-Perfect Branches · Dashed: Pruned Non-Credible Threats
Step Scrubber Highlights Backward-Induction Fold Order
Chapter §5 · Cooperative Games & Mechanism Design
Shapley (1953) · Gillies (1959) · Vickrey (1961)
The Shapley Value, The Core & Auction Revenue Equivalence
“The value of a game to player i is the average marginal contribution i makes across all n! possible arrival orders of the grand coalition.” — Lloyd S. Shapley (1953)
Theorem 5.1 (Shapley Axiomatization & Vickrey Bid Shading)
[Shapley 1953; Vickrey 1961]
Part A (Coalitions): ϕᵢ(v) = ∑S ⊆ N\{i} [|S|!(n−|S|−1)! / n!] · [v(S ∪ {i}) − v(S)] is the unique division satisfying Efficiency, Symmetry, Dummy, and Additivity. Yet ϕ(v) need not lie inside the Core {x : ∑i∈S xᵢ ≥ v(S)} unless (N, v) is convex.
Part B (Auctions): In a First-Price Sealed-Bid auction with n bidders and CRRA risk aversion r ∈ [0, 1), optimal bid shading is b*FP(v) = [(n − 1)/(n − r)]v, whereas Second-Price (Vickrey) auctions remain dominant-strategy truthful (b*SP(v) = v).
Part B (Auctions): In a First-Price Sealed-Bid auction with n bidders and CRRA risk aversion r ∈ [0, 1), optimal bid shading is b*FP(v) = [(n − 1)/(n − r)]v, whereas Second-Price (Vickrey) auctions remain dominant-strategy truthful (b*SP(v) = v).
ϕ₁ + ϕ₂ + ϕ₃ = v({1,2,3}) and b*_FP(v) = [(n − 1)/(n − r)] v ∎
Equation (5.1) — Live Shapley Permutations & Auction Bid Shading
Executable Formula
ϕᵢ(v) = (1/6) ∑_{π ∈ S₃} [ v(Pᵢ^π ∪ {i}) − v(Pᵢ^π) ], Core: xᵢ + xⱼ ≥ v({i,j})
The Counterfactual Twist · When the Core Collapses & Why Risk Aversion Breaks Revenue Equivalence
Shapley Value ϕ(v)(3.33, 8.33, 18.33)
Core Status & ϕ ∈ CoreNon-Empty · ϕ Inside
1st-Price Bid b*_FP(vᵢ)60.00 (25% Shade)
Vickrey 2nd-Price Bid80.00 (Truthful)
Fig 5.1 — Left: Barycentric Imputation Simplex, Shaded Core Polytope & Shapley Value Pin ◆ · Right: First-Price vs. Vickrey vs. All-Pay Bidding Curves
Shaded Region: Core · Brass ◆: Shapley Allocation ϕ(v)
| Arrival Order π ∈ S₃ | Marginal Δ₁(π) | Marginal Δ₂(π) | Marginal Δ₃(π) | Grand Sum v(N) |
|---|
Chapter §6 · Counterintuitive Equilibria & Field Index
Braess (1968) · Akerlof (1970) · Shubik (1971)
Braess's Network Paradox & The Counterfactual Compendium
“Adding an extra zero-cost link to a congested road network can strictly increase the travel time of every single driver at Nash equilibrium.” — Dietrich Braess, Unternehmensforschung (1968)
Theorem 6.1 (Braess's Paradox & The Price of Anarchy)
[Braess 1968; Roughgarden & Tardos 2002]
In a diamond routing network where N = 4,000 drivers travel from O to D via congestible links t(f) = f/100 and constant links t = 45, opening a zero-cost shortcut A → B makes the zig-zag path O → A → B → D a strictly dominant strategy for every individual driver — shifting the Wardrop Nash Equilibrium from 65 minutes to 80 minutes (PoA = 80/65 ≈ 1.231).
t_Nash(Bridge Open) = 40 + 0 + 40 = 80 min > t_Nash(Closed) = 65 min ∎
Equation (6.1) — Wardrop User Equilibrium vs. Pigouvian Social Optimum
Executable Formula
t_OA(f) = f / 100, t_AD = 45, Δt_Paradox = t_Nash(Open) − t_Nash(Closed)
The Counterfactual Twist · Why Closing Roads in Seoul, Stuttgart & NYC Sped Up Traffic
Nash Commute Time80.0 min
No-Bridge Baseline65.0 min
Paradox Penalty Δt+15.0 min Worse!
Price of Anarchy1.231× Social Opt
Fig 6.1 — Braess Diamond Network (O → A → D vs. O → B → D with Bridge A → B) · Edge Width Proportional to Equilibrium Driver Flow
Carmine: Congested Link · Prussian: Constant Link
Marginalia & Experiment Ledger (Archival Proof Log)
Stamped experiments across §1–§6 are recorded below with their exact parameter vector, classical theorem prediction, and perturbed outcome.
| Stamp # | Chapter | Game / Preset | Parameters & Perturbation | Classical Theorem vs. Simulated Twist |
|---|---|---|---|---|
| No experiments stamped yet — click "Stamp to Margin Ledger" in any chapter apparatus above. | ||||