Study notes on game theory (IV.) The best strategy of football match and business cooperation

Penalty cases

In a football match penalty, the penalty player can choose to l,m,r three different shooting paths, the goalkeeper can choose to jump to the left or to the left (in principle, he can also keep on the left).

	L	R
L	4,-4	9,-9
M	6,-6	6,-6
R	9,-9	4,-4

The table represents the respective benefits, where LR corresponds to 9 indicating that when the shooter shoots left and the goalkeeper jumps to the left, the shooter has a 90% chance to score, 9 means the goalkeeper has a 90% chance of losing the ball (10% probability of ejection). Other benefits and so on. We assume that the probability of the goalkeeper jumping to the left is Pr, then the goalkeeper jumps to the left PL=1-PR. So, the expected return on the left of the shooter is EU1(L,PR) = Pl*u1(l,l) + pr*u1(l,l) = (1-PR) + pr*9 = 4 + 5*PR; the expected revenue from the middle of the selection is eu< C3>1(M,PR) = Pl*u1(l,l) + pr*u1(l,l) = (1-PR) *6 + pr*6 = 6; the expected return on the on-the-off is EU1(R,PR) = Pl*u  1(l,l) + pr*u1(l,l) = (1-PR) *9 + pr*4 = 9-5*PR;

Conclusion: shooting from the middle is not the best strategy; Do not choose a strategy that is not the best strategy under any belief. 

Definition: The action of participant I si is the best countermeasure of the opponent's strategy S-I, when and only if for all other strategies of participant I si ', U1(s I, s-i) >=u1(Si ',S-i)

Business cooperation case

All two participants were shareholders of the company, who held shares in the company and split the profits.
SIRepresents the energy paid by the first shareholder for the company. i=1,2.
Total revenue is1+ S2+ b*s1*s2)
So for each participant, the benefit they can get is 1/2*4* (s1+ S2+ b*s1*s2) = (s)1+ S2+ b*s1*s2)
We now consider participant 1 and his pay is s1^2,s so his net income was: (s)1+ S2+ b*s1*s2)-S1^2
In order to make the most profit, the s1Derivation of the equation with a derivative of 0: s1= 1 + b*s2
In the same vein, for S2, S2= 1 + b*s1
We set up B=1/4 here. s=[1,4].

See here, because the range of S1 is only between 1 and 2, so [0,1] and [3,4] are the disadvantage strategies of s1 ; Similarly, [0,1] and [3,4] are the disadvantage strategies of s2 . So after culling left the S1∈[1,2],s2∈[1,2] This interval, we magnified it four times times, found the same diagram as the original. Then we can get the reception and get rid of it. The last point to get is the equation set: s1 = 1 + b*s2
s2 = 1 + b*s1
The solution. Draw: s1 = s2 = 1/(B-1) (1/(B-1), 1/(B-1)) This point is called  Nash equilibrium Nash equilibrium

This means that neither side of the game wants to deviate from the Nash equilibrium point. At the Nash equilibrium point, both sides take the best of each other's measures.

Study notes on game theory (IV.) The best strategy of football match and business cooperation