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L .A. Garcıa-Cortes et al. Livestock Production Science 64 2000 193 –202
195
The game proposed here is a n-person zero-sum the determination of the expected number of selected
game, that is, if a company increases its proportion sires coming from each company stock, N 5 p S .
i i
i
of the market, the proportion taken by other com- Order statistics for the whole set of normal dis-
panies will decrease by the same amount. In games tributions are not available, but given the distribution
of this kind, there is no reason for coalition, and the of the selection indexes, the truncation point that
only criterion for the ith company to optimize its satisfies N 2
op S 5 0 can be obtained by using the
i i
strategy n is to maximize N or p 5 N S . bisection algorithm Burden and Faires, 1985. This
i i
i i
i
Depending on the family size used by each algorithm evaluates the candidate points and, by
company, the expected number of sires selected from successive bracketing, approaches the truncation
k k
each company stock can be predicted. point k which satisfies N 2
op S 5 0. p is the
i i
i
proportion of sires selected from the ith company and the candidate truncation point k. After the
2.2. Game payoffs common truncation point was calculated, to obtain
the expected number of sires selected from each Each possible set of strategies n for all the players
i
company stock and the payoffs for each company is corresponds to a set of payoffs. A payoff is a number
straightforward from N 5 p S . that determines the winner of the game.
i i
i
Note that each company can calculate its payoff When playing this game, the companies will try to
only if it knows the strategies of the opponent maximize the expected number of sires selected from
companies, but the other players will hide their their stocks. So, in this case, a company will
strategies like in a gambling card game. To optimize consider itself as the winner of the game when the
its own strategy, each company should guess which proportion of selected sires which come from its
would be the strategies of the opponents. Further, to stock is bigger than the proportion its testing
play rationally the game, each company will assume facilities are of the total. For that reason, we define
that the other companies will also choose their the payoffs as a set of numbers derived from N :
i
strategies rationally. NT
i
] P 5 N 2
i i
T
3. Results: the analysis of the game
The payoffs can be determined after each com- pany has decided the strategyn . When the payoff of
i
3.1. Result of the game involving two companies: a company is zero, its expected proportion of the
the duopoly selected animals is equal to its proportion of the
testing facilities, N N 5 T T. A positive payoff for
i i
To illustrate the procedure that a company should the ith company means that the proportion of select-
follow to decide its strategy, we present in this ed sires coming from its stock is bigger than its
section the result of the game in a case with two proportion of the testing facilities. We show in this
companies. As an example, suppose that the testing section how to calculate the whole set of payoffs
facilities of the company 1 are 1000 spaces, the from the whole set of strategies.
testing facilities of company 2 are 500 spaces and the Setting the additive genetic variance equal to 1,
heritability is 0.25. They compete in a market that the stock of the ith company corresponds to S sires
i
will select 10 sires on the basis of the predicted ˆ
whose predicted additive values a are realized breeding value. Company 1 expects two thirds
samplings from a normal distribution with null 1000 1500 of the selected sires to come from its
expectation and variance equal to stock, and company 2 expects one third 500 1500.
2
In this game, both companies will calculate all 0.25n h
i 2
]]]]] s 5
ˆ a
2
possible pairs of strategies in order to decide its 1 1 0.25 n 2 1 h
s d
i
optimum family size. Table 1 presents a set of pairs Calculating the payoffs for each company requires
of pure strategies, from 2 to 16 progeny per sire for
´ ´
196 L
.A. Garcıa-Cortes et al. Livestock Production Science 64 2000 193 –202 Table 1
Payoff matrix for company 1 in the duopoly game; payoffs for company 2 are equivalent but opposite in sign. Strategies of company 1 are
a
presented in columns and strategies of company 2 are presented in rows n 52
n 54 n 56
n 58 n 510
n 512 n 514
n 516
1 1
1 1
1 1
1 1
n 52 1.591
1.942 1.980
1.884 1.713
1.494 1.246
2
n 54 21.745
0.494 0.609
0.563 0.434
0.260 0.061
2
n 56 22.087
20.487 0.132
0.110 0.007
20.139 20.308
2
n 58 22.014
20.578 20.128
20.016 20.106
20.237 20.391
2
n 510 21.796
20.509 20.101
0.015 20.084
20.206 20.351
2
n 512 21.529
20.375 20.006
0.097 0.080
20.117 20.254
2
n 514 21.255
20.215 0.118
0.209 0.190
0.112 20.132
2
n 516 20.990
20.049 0.253
0.333 0.312
0.235 0.127
2 a
The column and row presented in italic font correspond to the best payoffs for company 1 and company 2, respectively.
each company. This table is usually called the table eight progeny per sire and it contains the best
of payoffs and it can be calculated for each player payoffs for company 2.
before the game. Only the payoffs for company 1 are Both companies can use the approximation of
presented because company 2 has the same payoffs Robertson 1957 or numerical procedures to calcu-
but opposite in sign. Thus, the objective of company late that 14 progeny per sire maximizes the genetic
2
2 is to maximize its payoff or to minimize the payoff progress with T 51500, N 510 and h 50.25
of company 1.
0.5
T ]]
n 5 0.56 5 13.7
Fig. 1 shows the surface of the values presented in
S D
2
Nh Table 1, with the strategies ranging 2 to 20. It can be
shown that the opposing interests of the two com- So, both companies should decide to test their
panies have an equilibrium point that corresponds to sires using 8 daughters per sire the commercial
the row and column presented in bold in Table 1. In equilibrium or 14 daughters per sire the genetic
this table, the fourth column corresponds to eight optimum. Table 2 presents the relevant information
progeny per sire and it contains the best payoffs for that both companies will have available to decide
company 1, and the fourth row corresponds also to just between these two strategies. From Table 2, if
company 1 guesses that company 2 will choose 8, its payoff is better if it choose 8 0 over 20.237. In the
same way, if it guesses that company 2 will choose 14, its payoff is still better 0.209 over 0 by
choosing 8. Hence, company 1 will choose to test each sire with eight progeny, whatever the strategy
of company 2. The same argument holds for com- pany 2 to choose 8 progeny per sire. Hence, the
equilibrium point is unique and stable. Both com- panies have the same payoffs at 8,8 as at 14,14,
Table 2 Simultaneous strategies in the game involving two companies,
their payoffs and the total genetic progress of the population n
n N
N P
P EDG
1 2
1 2
1 2
8 14
6.876 3.124
0.209 20.209
1.215 14
8 6.430
3.570 20.237
0.237 1.229
8 8
6.667 3.333
1.201 14
14 6.667
3.333 1.242
Fig. 1. Payoffs for company 1 in the duopoly game.
´ ´
L .A. Garcıa-Cortes et al. Livestock Production Science 64 2000 193 –202
197
but they will choose 8 to prevent the competitive algorithm reaches the equilibrium and the set of
behavior of the opponent giving it an advantage. strategies is a Nash equilibrium. Although not pre-
In the theory of games framework, this kind of sented in this paper, we corroborated the uniqueness
stable optimum is called the Nash equilibrium Nash, of the Nash equilibrium by setting different sets of
1951. A set of pure strategies is a Nash equilibrium starting strategies. Each company considers fixed the
if no player can increase its payoff by a unilateral strategies of the opponents as fixed just as an
change in his strategy. The Nash equilibrium for algorithmic artifact to reach the Nash equilibrium.
games whose strategies have to be optimized within Usually, each company maximizes its payoffs by
a range of numerical values is usually called the differentiating its univariate payoffs function consid-
equilibrium of Cournot–Nash Binmore, 1992. In ering the strategies of the other companies fixed. In
general, this equilibrium does not have to be unique, this case, this function of payoffs is difficult to
but Fig. 1 shows a saddle form and the Nash differentiate and we used a successive bracketing
equilibrium is unique for this game. algorithm Press et al., 1986 to find the maximum.
So, the result of the game is that both companies This algorithm requires only several evaluations of
will simultaneously set their family sizes below the the payoff function to find its maximum.
2
genetic optimum and no company will win the game, We considered an example with N 510 and h 5
but the genetic progress will decrease from 1.242 to 0.25. The players were five companies whose testing
1.201. Under the assumptions made for this game, facilities were 700, 400, 200, 100 and 100, respec-
big companies do not win over small companies nor tively. As in the duopoly game, all companies have
vice versa, but companies that use the theory of to decide the optimum strategy and they have to
games will win over altruist companies that use the predict the optimum strategies of the opponents.
expected genetic progress as criterion. Starting strategies were arbitrarily set as 5 progeny
If only one of the companies decides unilaterally per sire for all companies. We show in case 1 of
to be competitive, Table 2 shows that company 2 Table 3 the result of the algorithm that converged in
will increase its expected number of successful sires 2 rounds of iteration. The 3rd round was calculated
from 3.33 to 3.57 7, and company 1 will increase to verify that results do not change and hence that
its results from 6.66 to 6.88 3. That is, the small the set of strategies reached is the Nash equilibrium.
company will be more interested than the big The algorithm presented in case 1 of Table 3 can
company to be competitive. In these circumstances, be interpreted as follows. First, all players can
it can be noticed that a previous agreement between calculate that company 1 will choose n 58.299 as
1
companies to be simultaneously altruist is difficult, the optimum strategy using the successive bracketing
as in any other zero-sum game. algorithm on its payoff function and considering that
the fixed strategies of the other companies were 3.2. Result of the game involving more than two
n 55, n 55, n 55 and n 55. Afterwards, all
2 3
4 5
companies players know that company 2 will guess the strategy
of company 1 and it will choose n 58.461 as the
2
The game involving two companies was analyzed optimum strategy considering n 58.299, n 55,
1 3
by setting several pairs of strategies and looking at n 55 and n 55. Following this successive set of
4 5
the payoff surface Fig. 1. If more than two optimizations where each company can predict the
companies are playing the game, the number of behavior of the other companies, the algorithm
possible sets of strategies grows dramatically and the reaches the Nash equilibrium. Note that the family
multidimensional table of payoffs is difficult to size was not forced to be an integer number, as in the
represent. For that reason, each company uses nu- duopoly game.
merical procedures to obtain the Nash equilibrium. Table 3 also shows four other cases with different
The algorithm starts from an arbitrary set of number of companies and different testing facilities.
strategies, and each company successively calculates All cases studied considered T 51500. From this
its optimum considering fixed the strategies of the table, we can conclude that the commercial optimum
opponents. After several rounds of iteration, the is equivalent for all the companies and does not
´ ´
198 L
.A. Garcıa-Cortes et al. Livestock Production Science 64 2000 193 –202 Table 3
Results and convergence of the algorithm reaching the Nash equilibrium in several cases with 2, 3, 4 or 5 companies and several testing
a
facility structures Case
T T
T T
T Iter
n n
n n
n
1 2
3 4
5 1
2 3
4 5
1 700
400 200
100 100
1 8.299
8.461 8.551
8.598 8.647
2 8.649
8.649 8.649
8.649 8.649
3 8.649
8.649 8.649
8.649 8.649
2 600
300 300
150 150
1 8.261
8.377 8.505
8.574 8.649
2 8.649
8.649 8.649
8.649 8.649
3 8.649
8.649 8.649
8.649 8.649
3 850
350 250
50 1
8.357 8.506
8.623 8.648
2 8.649
8.649 8.649
8.649 3
8.649 8.649
8.649 8.649
4 900
500 100
1 8.378
8.599 8.648
2 8.649
8.649 8.649
3 8.649
8.649 8.649
5 900
600 1
8.378 8.648
2 8.649
8.649 1
8.649 8.649
a
Heritability was 0.25 and the number of sires selected by truncation was 10. T are the testing facilities of each company and n is the
i i
strategy of each company.
depend on the number of companies involved in the the ratio T N on the family size and the expected
game. Moreover, it does not depend on the propor- genetic progress. With N fixed at 10 sires to be
tion of testing facilities of each company. In fact, it selected, the ratio T N ranged from 20 to 500 and
depends only on the ratio T N and the heritability. In heritability was 0.20. These figures show that the
the next section, we will investigate the relationship family size determined in order to cooperate is also
between the heritability, the ratio T N, the commer- larger than the family size to compete, whatever the
cial optimum and the genetic optimum. ratio T N. All cases show that competition produces
a decrease in the expected genetic progress. 3.3. Effect of the heritability coefficient and the
ratio T N on the commercial equilibrium
4. Discussion
