Saddlepoint — Interactive Game Theory Monograph & Counterfactual Laboratory

RAND SITUATION ROOM · 1962

Tactical Strategy Arcade — 5-Boss Campaign

YOU · RAND Field Commander Morale: 100 / 100 HP

Campaign Score: 0 pts Combo: 1.0× Spy Intel: 3 / 3 Envelopes Cleared: 0 / 5 Bosses

Bout #1 · Stakes

1× STANDARD

Boss Envelope: [CLASSIFIED]

Richard Thaler · Fair-Split Behavioralist Boss HP: 100 / 100 HP

“Real humans punish greedy splits and spread troops evenly. Show me if you can exploit fairness without triggering mutiny.”

Click or drag directly on the War Table — or deal a Tactical Card below — then press FIRE SALVO Ready for Turn #1

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)

Apparatus 1.A — Interactive 2 × 2 Bimatrix & Best-Response Square

Canonical Textbook Game

Quantal Rationality λ 4.0

Fictitious Play Prior p₀ 0.25

Fictitious Play Prior q₀ 0.80

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)

Apparatus 2.A — Automaton Duel, Punchcard Tape & 8 × 8 Ecological Tournament

Automaton I (Row)

Automaton II (Column)

Temptation T 5.0

Reward R 3.0

Punishment P 1.0

Trembling-Hand Noise ε 0.05

Continuation δ 0.90

Horizon N & Seed 40 rds

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

Apparatus 3.A — Barycentric Simplex Phase Portrait & Click-to-Drop Orbits

Evolutionary Ecosystem

Mutation / Exploration μ 0.000

Initial Share x₁(0) 0.55

Initial Share x₂(0) 0.30

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

Apparatus 4.A — Interactive Extensive Tree & Backward-Induction Scrubber

Sequential Game Tree

Disadvantageous Envy α 0.00

Advantageous Guilt β 0.00

Induction Fold Step All Steps (Full SPNE)

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).

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

Apparatus 5.A — 3-Player Coalition Characteristic Function v(S) & Auction Comparator

Coalition Game Preset

Coalition v({1,2}) 20

Coalition v({1,3}) 30

Coalition v({2,3}) 30

Grand Coalition v({1,2,3}) 30

Auction Bidders n 4

Bidder Risk Aversion r 0.00

Private Valuation vᵢ 80

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

Apparatus 6.A — Braess Diamond Network Flow & Searchable Paradox Catalog

Cross-Bridge A → B Status

Total Commuters N 4000

Bridge Delay / Pigouvian Toll c_AB 0 min

Filter Compendium Cards

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

Chapter §7 · RAND Wargame Table & Gauntlet Borel (1921) · Keynes (1936) · Schelling (1960)

Playable Strategic Gauntlet: Blotto, Level-k, Brinkmanship & Mutiny

“Strategy is not concerned with the efficient application of force, but with the exploitation of potential force — and the art of out-thinking an adversary who is trying to out-think you.” — Thomas C. Schelling (1960)

Theorem 7.1 (Borel–Gross–Wagner Colonel Blotto Simplex) [Borel 1921; Gross & Wagner 1950]

When two commanders distribute S = 100 regiments across K ≥ 3 battlefronts with weights (w₁, w₂, w₃), no pure troop allocation is a Nash equilibrium: any deterministic deployment (t₁, t₂, t₃) is strictly defeated by sacrificing your strongest front and outnumbering your other two fronts by +1 regiment. Equilibrium requires randomizing on the simplex with continuous marginal densities.

Equation (7.1) — Live Wargame Payoff & Best-Counter Proof Executable Formula

U_I(t, a) = Σ_{k=1..3} w_k · sgn(t_k − a_k), Σ_k t_k = Σ_k a_k = 100

Loading Wargame Gauntlet state...

The Counterfactual Twist · Why Textbook Nash Loses to Real Opponents

In multi-front allocation, guessing games, brinkmanship, and sequential bargaining, playing the textbook Nash equilibrium blindly can backfire when your opponent uses focal points, bounded Level-k depth, or spite. Switch between the 4 tabletop games on the right to test your instincts against 5 historical strategists.

Apparatus 7.A — Interactive Wargame Table & Adversary Gauntlet

Strategist Elo: 1500 Record: 0W – 0D – 0L Streak: 0 Gauntlet: 0 / 5 Masters

★ Blotto Commander ★ Level-k Mindreader ★ Schelling Nerve ★ 98-Coin Captain ★ Truel Survivor

5-Boss RAND Ladder: ① Thaler ② Nagel ③ Schelling ④ Selten ⑤ von Neumann

Seated Adversary Across the Table

Dossier: Randomizes across the exact Gross–Wagner / minimax support so no deterministic counter-strategy can exploit him.

Front α · Mountain Pass (4 pts) 45 reg

Front β · River Bridge (3 pts) 35 reg

Front γ · Rail Depot (2 pts) 20 reg

Your Theater Score4.0 pts

Adversary Score5.0 pts

Round VerdictReady to Deploy

Textbook BenchmarkCounter Beats Pure

Fig 7.1 — Three-Front Colonel Blotto Theater · Carmine: Your Regiments · Prussian: Adversary Regiments · Brass: Best Counter-Allocation Round #1

Wargame Dispatches & Round History Tape (Click Canvas or Fire to Log)

#	Mode	Adversary	Your Move	Table Resolution	Verdict / Elo
No dispatches logged yet — launch a salvo, seal a guess, or fire a Truel round above.

Marginalia & Experiment Ledger (Archival Proof Log)

Stamped experiments across §1–§7 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.