Strategic Competition & Game Theory
Fudenberg-Tirole Top Dog & Puppy Dog Strategy Formula
Classify Stage-1 strategic over- vs under-investment into Drew Fudenberg and Jean Tirole’s (1984) four animal strategies (Top Dog, Puppy Dog, Fat Cat, Lean and Hungry) using Tough/Soft and Substitutes/Complements derivatives.
Why cost-reducing R&D calls for aggressive "Top Dog" over-investment in Cournot — yet cautious "Puppy Dog" restraint in Bertrand!
Drew Fudenberg and Jean Tirole (1984) and Bulow, Geanakoplos, and Klemperer (1985) unified industrial-organization strategy around two partial derivatives! Let Toughness T = dPi_2 / dK_1 measure how Incumbent 1’s Stage-1 investment K_1 directly affects Rival 2’s profit (T < 0 = Investment makes Incumbent TOUGH, e.g., R&D lowering marginal cost; T > 0 = Investment makes Incumbent SOFT, e.g., building loyal captive buyers that motivate higher prices), and let Slope S = R’_2 = dx_2* / dx_1 measure Stage-2 reaction slope (S < 0 = Strategic Substitutes like Cournot quantities; S > 0 = Strategic Complements like differentiated Bertrand prices)! Under Entry Deterrence (Goal = 1), Incumbent wants to push Pi_2 down: Tough (T < 0) -> Overinvest ("Top Dog") and Soft (T > 0) -> Underinvest ("Lean and Hungry")! Under Post-Entry Accommodation (Goal = 0), Strategic Effect equals (dPi_1 / dx_2) * S * (dx_1 / dK_1), whose sign matches T * S: Tough + Substitutes -> Overinvest ("Top Dog"); Tough + Complements -> Underinvest ("Puppy Dog"); Soft + Complements -> Overinvest ("Fat Cat"); Soft + Substitutes -> Underinvest ("Lean and Hungry")!
Verify quantitatively on an Accommodation game (Entry_Deterrence_Goal = 0 in B2) with Cross_Profit_Effect dPi_1/dx_2 = +15.0 (Bertrand rival higher price helps Incumbent, B3), Stage1_Own_Action_Shift dx_1/dK_1 = -2.0 (cost-cutting R&D lowers Incumbent’s own price -> Tough T = -30 < 0, B4), and Rival_Reaction_Slope S = dx_2*/dx_1 = +0.40 (Strategic Complements > 0, B5): Total Indirect Strategic Marginal Effect is (+15.0) * (+0.40) * (-2.0) = -12.00 (< 0 — over-investing triggers an aggressive rival price cut that hurts Incumbent by -12.00/unit!), classifying the optimal posture as "Underinvest: Puppy-Dog Ploy"!
| Fudenberg-Tirole Regime (Accommodation Goal = 0) | Sign Conditions | Optimal Strategic Posture |
|---|---|---|
| Tough (T < 0) + Substitutes (S < 0) | T * S > 0 (Cournot R&D) | Overinvest: Top Dog |
| Tough (T < 0) + Complements (S > 0) | T * S < 0 (Bertrand R&D) | Underinvest: Puppy-Dog Ploy |
| Soft (T > 0) + Complements (S > 0) | T * S > 0 (Bertrand Loyalty) | Overinvest: Fat-Cat Effect |
| Soft (T > 0) + Substitutes (S < 0) | T * S < 0 (Cournot Loyalty) | Underinvest: Lean and Hungry |
Stage-1 Indirect Strategic Marginal Effect dPi_1_strat / dK_1 Formula
Decompose total Stage-1 investment incentive into Direct Cost/Demand Saving PLUS Strategic Term (dPi_1 / dx_2) * (dx_2* / dx_1) * (dx_1* / dK_1)!
=ROUND(B3 * B5 * B4, 2)Exact Fudenberg-Tirole Four-Animal Strategy Classifier Formula
Infer Toughness sign cleanly from Rival Cross-Effect T_sign = B3 * B4 (in symmetric games, if rival higher action helps Incumbent B3 > 0 and K_1 lowers own action B4 < 0, K_1 hurts Rival so T < 0; similarly in Cournot B3 < 0 and K_1 raises own output B4 > 0 so B3 * B4 < 0 = Tough!).
=LET(g_det, B2, dp_dx, B3, dx_dk, B4, r_slp, B5, IF(OR(AND(g_det <> 0, g_det <> 1), dp_dx = 0, dx_dk = 0, r_slp = 0), "Zero Strategic Effect", LET(t_sgn, dp_dx * dx_dk, IF(g_det = 1, IF(t_sgn < 0, "Overinvest: Top Dog", "Underinvest: Lean and Hungry"), IFS(AND(t_sgn < 0, r_slp < 0), "Overinvest: Top Dog", AND(t_sgn < 0, r_slp > 0), "Underinvest: Puppy-Dog Ploy", AND(t_sgn > 0, r_slp > 0), "Overinvest: Fat-Cat Effect", TRUE, "Underinvest: Lean and Hungry")))))Try the example
Use the original values or change them to test your understanding.