There are ten locations with respective value a1<…<a10. Player i (i=1,2) is endowed with ni sold

First, letu0027s consider the case where all $a_p$ are distinct. In this case, each player wants to

There are ten locations with respective value a1<…<a10....

There are ten locations with respective value $a_1<\ldots<a_{10}$. Player i $(i=1,2)$ is endowed with $n_i$ soldiers $\left(n_i<10\right)$ and must allocate them among the locations. To each particular location he can allocate no more than one soldier. The payoff at location $p$ is $a_p$ to the player whose soldier is soldier at p or no one has, in which case the payoff is zero to both. The total payoff is obtained by summing up local payoffs. Show that this game has a unique dominating strategy equilibrium. What if some of the $a_p$ coincide?

There are ten locations with respective value $a_1<\ldots<a_{10}$. Player i $(i=1,2)$ is endowed with $n_i$ soldiers $\left(n_i<10\right)$ and must allocate them among the locations. To each particular location he can allocate no more than one soldier. The payoff at location $p$ is $a_p$ to the player whose soldier is soldier at p or no one has, in which case the payoff is zero to both. The total payoff is obtained by summing up local payoffs.
Show that this game has a unique dominating strategy equilibrium. What if some of the $a_p$ coincide?

Key Concepts

Dominant Strategy Equilibrium

A dominant strategy equilibrium is a solution concept in game theory where each player's chosen strategy is the best response to any strategies the opponent might pick. In such an outcome, no player has an incentive to deviate regardless of the opponent’s actions. This is particularly significant in settings where the payoff is determined locally (e.g., each battlefield or location) because selecting the optimal allocation for every individual contest guarantees the overall best response.

Resource Allocation in Competitive Settings

The game is a variant of resource allocation contests, often referred to as Colonel Blotto games, where players distribute a limited number of units across several battlefields or targets. The challenge involves choosing how to allocate limited resources efficiently to win the most valuable locations, underscoring the need for strategic planning based on both the value of the targets and the constraints on the number of units available.

Prioritization Based on Ordered Values

When the targets are associated with strictly increasing values, an intuitive strategy emerges: each player should deploy their units to the highest value locations available. This ordering eliminates ambiguity in decision making since higher rewards are clearly preferable, thereby leading to a unique equilibrium in which the optimal strategy is to prioritize the most valuable locations first.

Impact of Indistinguishable Values

If some targets share the same value, the clear hierarchy used for prioritization becomes ambiguous. In such cases, allocating a unit to one of the equally valued targets instead of the other does not change the payoff. This indifference can result in multiple equilibria since the uniqueness of the best allocation strategy relies on the strict ordering of values. Hence, when ties in values occur, the equilibrium may cease to be unique.

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