Consider an allocation by government of a fixed prize Z between two groups of contenders. The groups of contenders may be regions or academic fields. The contenders may be firms, research institutions, local governments, or individuals. Total population of N contenders are divided into two groups denoted 1 and 2, respectively, where N ≥ 3. Group i consists of ni contenders, where 1 ≤
for i = 1, 2, and n1 + n2 = N. Without loss of generality, we assume that n1 ≥ n2.
The prize increases in the number of contenders. The more contenders participate in the contest, the larger pecuniary prize is given to the winner of the contest. Even when the pecuniary prize is fixed, social psychological prize such as honor and pride may accrue to the winner of the contest.
As in Katz and Tokatlidu [
, for i = 1, 2, even though the government allocates 
to group i.
The parameter g indicates the magnitude of size effect. In this case, g may represent non-pecuniary psychological prize such as honor or pride. Or g may be viewed as the magnitude of future expected benefit. As the more contenders can signal the potential quality of the winner, this may increase the possibility of sponsorship contracts in the future, for example. The parameter g belongs to the closed interval [0, 2] as in Eggert and Kolmar [
In order to analyze the two-stage contest, we solve backwards. That is, utilizing the results derived from the analysis of the second-stage of the contest, we later examine the first-stage inter-group share contest.
Risk-neutral contender j of group i expends an amount eij of resource in units commensurate with the prize, for i = 1, 2, and j = 1, 2,···, ni, in the second-stage intra-group contest. Specifically, contender j of group i solves the following problem, for i = 1, 2, and j = 1, 2,···, ni,
where vij denotes the expected payoff to contender j of group i and pij is given by 
Simultaneous solution of the first-order conditions for Equation (1) gives the equilibrium effort
, the individual payoff 
of each contender, and the group payoffs
, as follows:
and
for i = 1, 2. (2.3)
The parameter g does affect the individual payoff as well as the group payoffs.4 The payoffs to the contenders increase by the factor of 
for contenders in group i, for i = 1,2. Note that
, 
and 
all increase in g. Moreover, the larger the group, the greater is the size effect, if g > 0.
We now examine the first-stage share contest between the two groups. This paper considers two scenarios of endogenous shares. A natural scenario is that the contenders determine their contribution to the share contest by a non-cooperative Nash strategy, as they do in the second stage. Another scenario is that the contenders in each group cooperate in the share contest. In such a case the representative of each group chooses the level of contribution that maximizes the group payoffs. The shares are determined to be proportional to group contributions in the share contest.
We first consider the case when the contenders employ a non-cooperative Nash strategy. Contender j of group i contributes an amount xij in units commensurate with the prize to the share contest, for i = 1, 2, and 
We denote the group expenditures in the first stage by X1 and X2. That is, 
and
.
Group i’s share is assumed to be given by, for i = 1, 2,
where NC denotes the solution for the non-cooperative Nash regime. In the first-stage, contender j of group i solves the following maximization problem, for i = 1, 2, and
.
Simultaneous solution of the first-order conditions for maximization in the first stage yields the equilibrium group expenditures and the equilibrium shares in the first stage of the non-cooperative Nash regime, for i, j = 1, 2, and i ≠ j,
and
Equation (5) shows that the smaller group expends more than the larger group in the first stage, and as a result, obtains the larger share when the contenders employ a non-cooperative Nash strategy in the first-stage share contest. However, the size effect through prize mitigates the disadvantage of the larger group, to some extent. The share of the larger group, 
, increases in g. At the maximum of g = 2, the larger group, group 1, obtains the half of the prize.
Inserting 
into Equation (2), we obtain the equilibrium effort of each contender, the individual payoff and the group payoffs:
and
We now examine the inter-group share contest when the contenders in each group respectively cooperate to maximize the group payoffs. In such a case, the representative of group i solves the following maximization problem, for i = 1, 2,
Simultaneously solving the first-order conditions for Equation (7), we obtain the equilibrium group expenditures and the equilibrium shares in the cooperative regime, for i, j = 1, 2, and i ≠ j,
and
It is evident that the size effect increases the share of the larger group. If g = 1, the two groups obtain the equal share of the prize. Moreover, if 1 < g ≤ 2, then the larger group obtains the larger share. Inserting 
into Equation (2), we obtain the equilibrium effort in the second stage, the individual payoff and the group payoffs in the cooperative regime:
and
We now compare the non-cooperative Nash regime with the cooperative regime. Under the assumption that n1 ≥ n2 and N ≥ 3, we obtain:
1)
, where the equality holds when n1 = n2.
2)
.
3)
, where the equality holds when n1 = n2.
4) 
and
, where the equalities hold when n1 = n2.
The size effect through prizes mitigates, to some extent, the group size disadvantage resulting from fierce intra-group competition in the larger group. Moreover, if the size effects are strong enough in the cooperative regime, the larger group can even obtain the larger share. The groups expend more in the first-stage share contest in the cooperative regime than in the non-cooperative Nash regime. This is because the larger group is free of the free-rider problem in the cooperative regime.
The afore-mentioned results draw some important insights regarding government policies. In our set-up, how the contenders in each group perceive the psychological benefit of the size effect has crucial effect on the equilibrium outcomes. If contenders care more about psychological benefit, then the size effects become strong enough and different outcomes may follow. Moreover, different frames may lead to different motives and eventually may induce different equilibrium outcomes. In this case, the role of group leader may be important. For example, if an organization penalizes free riders severely, then the free rider problem is minimized within this organization. The contenders regard as if they are in a cooperative regime rather than in a non-cooperative Nash regime. Therefore, it is important how an organization (or its leader) delivers incentives (psychological benefits, especially) and sets up the frame for its members.