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Interpret the contests as standard all-pay auctions: the highest effort among entrants wins, ties are shared uniformly, and an unentered contest awards no prize. The printed question does not specify a prize allocation rule; the highest-effort convention is the one used for standard all-pay contests in the course author's 2014 lecture slides. Under this convention, we can characterize the unique symmetric participation probabilities and bid marginals. Uniqueness of the entire joint mixed strategy requires a further restriction on dependence, as explained below.

Let $q_j$ be the probability that a player omits contest $j$. Since every player enters exactly two contests, $q_1+q_2+q_3=1$. Let $G_j(b)$ be the probability that a rival is absent from contest $j$ or enters it with effort at most $b$. Rivals' strategy draws are independent between players. For a positive bid $b$ outside a measure atom, the expected payoff from this contest isAn all-pay auction cannot have a positive-effort measure atom in a symmetric equilibrium: slightly overbidding that measure atom gives a positive discrete increase in the winning probability at an arbitrarily small extra cost. Nor can active bids have a measure atom at zero when entry has positive probability, because a small positive bid beats tied zero bids. Gaps inside the active effort support are impossible: moving a bid from the top of a gap to just above its bottom preserves its winning probability and lowers its cost. The effort support starts at zero, because lowering its positive lower endpoint would preserve the chance that every rival is absent.

The all-pay indifference equation with random entry consequently gives the maximal per-contest payoffEvery $q_j$ lies strictly between zero and one. First, if $q_j=1$, the other two contests have certain entry and zero per-contest payoff, while deviating into the unused contest wins a positive prize. This contradicts equilibrium. Next, if $q_j=0$, the remaining omission probabilities sum to one and neither can equal one, so both are positive. Contest $j$ has zero per-contest payoff, whereas both other contests have strictly positive payoffs. Every pair containing $j$ is then worse than omitting it and entering the other two, contradicting certain entry in $j$.

Every omission therefore occurs with positive probability. The three entered pairs must give the same maximal payoff, which forces $u_1=u_2=u_3=u>0$. Normalizing the omission probabilities gives the two-of-three all-pay participation equilibrium:Conditional on entering contest $j$, the effort has distribution functionextended by zero below this interval and one above it. The inequalities $q_1\le q_2\le q_3$ show that the larger prizes are entered more often.

To construct an equilibrium, omit $j$ with probability $q_j$, then draw the two active efforts independently with their respective conditional distributions $H_k$. Every bid in a contest's effort support earns $u$; a bid above $w_j-u$ earns at most $w_j-b<u$. Thus no effort deviation or choice of a different pair improves on total payoff $2u$. This proves existence and verifies the Nash equilibrium without relying only on the indifference equations. The arguments above also prove uniqueness of the omission probabilities and the per-contest marginal distributions.

For the full joint strategy law, however, the printed uniqueness claim is too strong. Given an entered pair $k,\ell$, either use two independent uniform random variables $U,V$ and bids $(H_k^{-1}(U),H_{\ell }^{-1}(V))$, or use a single uniform $U$ and bids $(H_k^{-1}(U),H_{\ell }^{-1}(U))$. These are different copulas with the same conditional marginals. A fixed deviation's expected additive payoff only uses the rivals' per-contest marginal distributions, so both constructions remain Nash equilibria. This is the marginal-equivalent equilibria in additive contests phenomenon. The participation probabilities and bid marginals are unique; the full joint mixed strategy is not unique unless a dependence convention is imposed. Independent conditional sampling gives one canonical representative.