Game Theory & Strategic Coordination
Harsanyi-Selten Risk Dominance (Stag Hunt) Formula
Calculate the Critical Trust Threshold p* = (D - C)/((A - B) + (D - C)) and Harsanyi-Selten (1988) risk-dominance margin in symmetric 2x2 Stag Hunt coordination games.
Why teams get trapped in safe low-output equilibria even when everyone prefers the high-output outcome!
In Jean-Jacques Rousseau’s Stag Hunt coordination game (payoffs: mutual Stag = A, Stag while partner hunts Hare = C, Hare while partner hunts Stag = B, mutual Hare = D, with A > B and D > C), BOTH (Stag, Stag) and (Hare, Hare) are strict pure-strategy Nash equilibria! John Harsanyi and Reinhard Selten (1988) distinguished Payoff Dominance (A > D -> mutual Stag makes everyone better off) from Risk Dominance: Stag is best-response ONLY when perceived partner cooperation probability p exceeds Critical Trust Threshold p* = (D - C) / ((A - B) + (D - C))! Whenever p* > 0.50 (equivalently, Hare deviation loss D - C > Stag deviation loss A - B), safe (Hare, Hare) is Risk Dominant — and evolutionary/noisy learning converges onto (Hare, Hare)!
Verify on Stag_Stag_A = 10 (B2), Sucker_Stag_Hare_C = 0 (B3), Free_Hare_Stag_B = 8 (B4), and Safe_Hare_Hare_D = 7 (B5): (Stag, Stag) = 10 strictly payoff-dominates (Hare, Hare) = 7, YET Stag deviation loss is A - B = 2.00 whereas Hare deviation loss is D - C = 7.00, pushing Critical Trust Threshold up to p* = 7 / (2 + 7) = 0.7778 (77.78% > 50% -> Hare is Risk Dominant by +5.00!). Now lower solo Hare temptation B from 8 down to 2: Stag gain A - B rises to 8.00 > 7.00, dropping p* to 7 / 15 = 0.4667 < 0.50 so Stag becomes BOTH Payoff-Dominant AND Risk-Dominant!
| Stag Hunt Metric (A=10, C=0, B=8, D=7) | Formula | Verified Output |
|---|---|---|
| Stag Deviation Loss (Gain to Matching Stag) | A - B | 2.00 |
| Hare Deviation Loss (Safety Over Sucker C) | D - C | 7.00 |
| Critical Trust Threshold (p*) | (D - C) / ((A - B) + (D - C)) | 0.7778 (77.78% Trust Required!) |
| Harsanyi-Selten Risk-Dominance Regime | IF(p* < 0.5, "Stag", IF(p* > 0.5, "Hare", "Tie")) | Hare Risk-Dominant (p* > 0.50) |
Critical Cooperation Belief Threshold p* Formula
Validate strict coordination game ordering A > B and D > C so both (Stag, Stag) and (Hare, Hare) are genuine strict Nash equilibria.
=LET(a_ss, B2, c_sh, B3, b_hs, B4, d_hh, B5, IF(OR(a_ss <= b_hs, d_hh <= c_sh), "Not a Coordination Game (Require A > B and D > C)", ROUND((d_hh - c_sh) / ((a_ss - b_hs) + (d_hh - c_sh)), 4)))Harsanyi-Selten Risk-Dominant Equilibrium Selector Formula
Under 50/50 Laplace strategic uncertainty, Stag expected payoff 0.5*(A + C) beats Hare expected payoff 0.5*(B + D) iff p* < 0.50!
=LET(a_ss, B2, c_sh, B3, b_hs, B4, d_hh, B5, IF(OR(a_ss <= b_hs, d_hh <= c_sh), "Not a Coordination Game (Require A > B and D > C)", LET(m_st, (a_ss - b_hs) - (d_hh - c_sh), IF(m_st > 0, "Stag Risk-Dominant", IF(m_st < 0, "Hare Risk-Dominant", "Risk-Neutral Tie")))))Try the example
Use the original values or change them to test your understanding.