Livestock Production Science 64 2000 193–202 www.elsevier.com locate livprodsci
Optimum family size in progeny testing and the theory of games
´ ´
¨ L.A. Garcıa-Cortes , G. Yague, C. Moreno
´ Unidad de Genetica Cuantitativa y Mejora Animal
, Facultad de Veterinaria, Universidad de Zaragoza, c Miguel Servet, 177, Zaragoza, E-
50013, Spain Received 14 December 1998; received in revised form 26 May 1999; accepted 27 August 1999
Abstract
In this paper, the optimum family size in a progeny test with limited testing facilities was determined for a scheme where several commercial companies were competing. Companies which determined family size in order to maximize the expected
proportion of sires that will be selected from its stock were considered as competitive. On the other hand, companies that determined family size in order to maximize the expected genetic progress were considered as altruist. Using the theory of
games, it was shown that competitive companies obtain better commercial results than altruist companies. When competing against competitive companies, altruist companies obtained worse commercial results than they expected. When all
companies were competitive, the commercial results equalled those when all were altruist, but the total genetic progress decreased. A numerical procedure is described to calculate the family size to optimize the commercial results. The result of
this algorithm showed that this commercial equilibrium depends only on the heritability and the ratio between the total testing facilities of the population and the number of sires required for the market. This commercial equilibrium did not
depend on the number of companies or the size of each company.
2000 Elsevier Science B.V. All rights reserved.
Keywords : Animal breeding; Progeny test; Family size; Theory of games; Nash equilibrium
1. Introduction values as criteria. These plans require the additive
genetic values to be predicted as accurately as Animal breeding plans are usually based on
possible, but usually there is a conflict between the selecting by truncation the animals to be used as
number of parents that can be tested and the accura- sires and dams in the next generation. Parents are
cy of their genetic evaluations Robertson, 1957; selected by using their predicted additive genetic
James, 1979; Fernando and Gianola, 1990; Bourdon, 1997. In this context, designing a progeny test to
evaluate new sires with a fixed amount of testing facilities will require the breeder to determine how
many sires will be evaluated and thus how many
Corresponding author. Tel.: 134-76-761-000, ext. 4207; fax:
progeny will be used to test each of them. Although
134-76-761-612. ´
´ E-mail address
: agarcorposta.unizar.es L.A. Garcıa-Cortes
the accuracy of the evaluations can be used as a
0301-6226 00 – see front matter
2000 Elsevier Science B.V. All rights reserved. P I I : S 0 3 0 1 - 6 2 2 6 9 9 0 0 1 4 8 - 7
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194 L
.A. Garcıa-Cortes et al. Livestock Production Science 64 2000 193 –202
criterion to optimize the programs Foulley et al., 2. Methods
1983, the optimum number of progeny per sire is usually analyzed in terms of the expected genetic
In this section we propose a game where a given response.
number of companies the players design simul- When several commercial companies design their
taneously progeny tests for new artificial insemina- progeny tests, they will have to increase the genetic
tion AI sires. Although Ashtiani and James 1993 merit of their breeding stocks in order to be competi-
have shown that considering different strains can tive. Nevertheless, the objective of commercial com-
modify the optimum number of progeny per sire, we panies is to increase the proportion of the market that
will assume that animals coming from different they can have in terms of selected AI sires or
company stocks are sampled from the same popula- selected animals. Usually, the relationship between
tion. the genetic merit of the breeding stock and the
proportion of the market is difficult to analyze Hill, 2.1. The rules and the notation
1971. Although the companies will try to increase the genetic merit of their stocks to be competitive,
Each company has a fixed amount of testing they will design their breeding plans in order to
facilities, that is, a fixed number of progeny that can increase their benefits and not to increase the genetic
be measured. Hence, the number of sires that each progress explicitly.
company can test depends on the number of progeny The genetic progress is expected to be optimum
used to test each sire, and we will call the number of when companies try to optimize it explicitly. Never-
progeny per sire family size. All companies know
2
theless, when companies are competitive the genetic the heritability h , the number of tested AI sires
progress will be an indirect effect of the competition. demanded for the market and the amount of testing
The intuition that guided our search is that the facilities of the opponents. The game will be played
genetic progress obtained as an indirect result of the for only one generation.
competition will be smaller than the optimum. The Companies can use their own family size and
theory of games von Neumann and Morgenstern, afterwards the market will choose the sires with the
1944 provides a formal approach to the analysis of best selection index within the whole set of tested
the optimal decision of the companies, or players, in sires. The objective of each company is to maximize
cases where their interests are interdependent. the number of selected sires coming from its stock.
The objective of this paper is to investigate, using We will use the following notation: T will be the
the theory of games, the impact of this competitive total amount of testing facilities, N will be the
behavior. We will compare the genetic progress number of tested sires which the market will require
obtained in a scheme whose companies try to after the testing period and n will be the family size
increase their proportion of the market with another used for testing the sires. Corresponding to this
scheme whose companies try to increase the genetic notation, T will be the amount of testing facilities of
i
merit of their stocks. We will also analyze how to the ith company, n will be the family size that the
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calculate the optimal decision of each company in ith company uses for testing its sires, S 5 T n will
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order to maximize the number of parents selected be the number of sires tested for the ith company and
from its stock. p 5 N S is the proportion of sires selected from
i i
i
In this paper, we propose a simple game that will those tested.
try to catch the essentials of the competition between Fortunately, the analysis of this game under the
companies. We analyze the simplest case of the theory of games point of view is simple. For that
game, where only two companies are competing as a reason we do not include an introduction about the
duopoly and we generalize the game to include theory of games here. From here on, we will outline
several companies. Finally, we analyze the effect of definitions and explanations to illustrate the analysis
the heritability and the amount of testing facilities on where necessary. A general introduction about the
the result of the game. theory of games can be found in Rasmusen 1994.
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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 .
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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
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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.
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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
