THE COACH’S DILEMMA by Daniele & Gianfranco Gambarelli

Group D FIRST DAY: June 14th Costa Rica Uruguay 3-1 Italy England 2-1

Group D COSTA RICA 3 ITALY ENGLAND URUGUAY

NEXT MATCH Italy Costa Rica vs

Cesare Prandelli Jorge Luis Pinto ATTACK OR DEFENSE? Costa Rica Italy Cesare Prandelli Jorge Luis Pinto

THE FORECASTS

, , THE FORECASTS THE PAYOFFS 0.56 x 3 + 0.34 x 1 = 2.02 DEFENSE ATTACK , , % ITA win. draw C.R. 56, 34, 10 76, 19, 05 61, 24, 15 66, 09, 25 66, 14, 20 THE PAYOFFS DEFENSE - DEFENSE ITA Payoff 0.56 x 3 + 0.34 x 1 = 2.02 C.R. Payoff 0.10 x 3 + 0.34 x 1 = 0.64

, THE STRATEGIES THE PAYOFFS THE SOLUTION < < < < ITA Payoff C.R. Payoff DEFENSE ATTACK < 2.02 , 0.64 2.47 , 0.34 < < < 2.07 , 0.84 2.12 , 0.74 THE SOLUTION ATTACK DEFENSE

NO DOMINANCE? Nash (1950) Having called qAD = % defense of A qAO = % attack of A h = (aOD - aOO) / (aDO + aOD - aDD - aOO), it is: qAD = h if 0  h  1. Elsewhere  0 if min (aDD, aDO) < min (aOD, aOO) qAD =  1 if min (aDD, aDO) > min (aOD, aOO)  x: 0 < x < 1 otherwise qAO = 1 - qAD.

THE STABILITY +10% -10% DEFENSE ATTACK 61, 65, 15 20, 31, 08 61, 65, 15 20, 31, 08 20, 24, 15 73, 65, 03 56, 34, 10 76, 19, 05 71, 65, 15 20, 08, 21 68, 17, 15 20, 65, 15 66, 09, 25 66, 14, 20

AN OPEN PROBLEM: Discrete  Continuum

CONCLUSIONS: … To Game-Theorists: To Coaches: To Curious: Discrete  Continuum To Coaches: Before the match  No stone unturned To Curious: …

What happened?

but …

ITALIA 0 – COSTA RICA 1