2 More on Oneout Twobase baseball The key strategic variable

2) More on One-out, Two-base baseball. The key strategic variable for the team in the field is where to position the rover fielder. The key strategic variable for the batting team is where to hit the ball. The following table gives the strategies and payoffs for the first play. The number in the table is the probability of a hit. The team at bat wants a large probability, the team in the field wants a small one. Experience has taught the rovers to wait at the mound and then move suddenly to the left, right or center field as the ball is hit. Thus, you can think of this as a simultaneous move situation. Team in the field Right fieldCenter field ft field mound Bunt 250 150 180 100 Hit to right 240 280 300 230 Team at bat Hit to left 400 300 150 220 Hit to center 350 350 210

Solution

The probabilities of a hit P(hit) have been specified in the question and have been stated in the table below in bold. The probability of placing the fielder will thus be (1-P(hit)) which have been written in normal font. It has also been mentioned that the row player or team at bat looks for a higher probability and the team at field who is the column player looks for a lower probability. Therefore, the table of payoffs can be completed as presented in Table 1 below:

a) A dominant strategy is one where regardless of what the other player chooses, the strategy earns the player in question the larger (or smaller for negative payoff) payoff. Here, since the team at bat is the row player, the payoffs in bold indicate the team at bat’s payoff. When the team at bat chooses “hit to centre”, the payoffs strictly dominate those of the strategy “Bunt”. This is because, for each choice, 0.350>0.250, 0.200>0.150, 0.350>0.180 and 0.210>0.100. Thus, the row “Bunt” gets eliminated and the row “Hit to Center” becomes the dominant strategy.

b) Following the same logic in part a), since “Hit to Centre” dominates “Bunt”, “Bunt” becomes the dominated strategy.

c) The responses of each team based on the opponent’s moves are as given in the table 2 below. The best response of team at bat requires choosing the highest payoff for each strategy of the team at filed and the best response of team at filed requires choosing the lowest payoff for each strategy of the team at bat.

d) Nash equilibrium is a stable equilibrium wherein no player has any incentive to deviate from the chosen strategy given that the strategy of the opponent remains unchanged. In this case, when the team at bat chooses “hit to Left”, the team at field chooses “Right Field” and when the team at field chooses “Right Field”, the team at bat also chooses “Hit to Left”. Thus, (Hit to Left, Right Field) is one Nash equilibrium. Similarly when the team at bat chooses “Hit to Center” the team at Field can choose “Left Field” and when the team at Field chooses “Left Field” the team at Bat chooses “Hit to Center”. Thus (Hit to Center, Left Field) is another Nash equilibrium.

TABLE 2

If team at Bat does...

Then best response for team at field is to stay at...

If team at filed does...

Then best response for team at bat...

Bunt

Right Field (0.750 is the lowest payoff)

Right Field

Hit to Left (0.400 is the highest payoff)

Hit to Right

Left Field

Center Field

Hit to Left

Hit to Left

Right Field

Left Field

Hit to Center

Hit to Center

Either at Right Field or Left Field

Pitcher’s mound

Hit to Right

TABLE 1

Right Field

Center Field

Left Field

Pitcher’s mound

Bunt

0.250, 0.750

0.150, 0.850

0.180, 0.820

0.100, 0.900

Hit to Right

0.240, 0.760

0.280, 0.720

0.300, 0.700

0.230, 0.770

Hit to Left

0.400, 0.600

0.300, 0.700

0.150, 0.850

0.220, 0.780

Hit to Center

0.350, 0.650

0.200, 0.800

0.350, 0.650

0.210, 0.790

If team at Bat does...	Then best response for team at field is to stay at...		If team at filed does...	Then best response for team at bat...
Bunt	Right Field (0.750 is the lowest payoff)		Right Field	Hit to Left (0.400 is the highest payoff)
Hit to Right	Left Field		Center Field	Hit to Left
Hit to Left	Right Field		Left Field	Hit to Center
Hit to Center	Either at Right Field or Left Field		Pitcher’s mound	Hit to Right