Consider a person playing a game or making a decision in a strategic situation. If several people are involved then the choices made by one player may influence the choices of others. By making the right choices, each player hopes to gain some kind of reward or payoff. Game Theory is the branch of mathematics that analyzes the decision-making strategies of people to understand the possible outcomes of a strategic situation. A subfield of game theory is called Games on Graphs. Problems in this field typically involve a collection of players who are connected by a network. For example, if the players are human beings, then the network might consist of social connections. In a game, these connections become important as they determine whose choices each player can “see.” As the game progresses, each player is only aware of the choices made by players that he or she is connected to (as opposed to the choices made by all players in the game). Researchers have demonstrated that the structure of the social network can have a dramatic impact on the outcome of the game (Anderlini & Ianni, 1996). Games can be classified according to the conditions under which players succeed. For example, in coordination games (Schelling, 1960) players win by making the same choices. A recently identified type of game is the spatial dispersion game (Alpern, 2001). In a spatial dispersion game, the positions of different players are often represented as a location (x, y) on a plane. Players may achieve payoffs if they spread themselves evenly (e.g. territorial animals colonizing a small island) or if they congregate near a desirable spot, depending on the rules of the game. Imagine a game in which each player has to select a location (x, y) on a two dimensional plane. Players may occupy the same location if they wish, and can modify their choice of location by observing the choices of others. When all of the players have selected their locations, we can calculate a mean location by averaging. If we give the greatest payoff to the player closest to the mean, then the game is a two-dimensional beauty contest with multiple players (BoschDomenrech et al., 2002). Imagine a more complicated beauty contest in which the mean position is still desirable, but in which each player wishes to express individuality. When the players have made their choices we will be able to determine the most popular (or mode) location. In this game, the greatest payoffs go to those players who are able to choose locations that are close to the mean and far from the mode position. We propose to study multidimensional versions of this game. We will begin by generalizing the rules of the game from two (x, y) to an arbitrary number of dimensions (x1,…, xn). Instead of representing locations in space, the points (x1,…, xn) will represent each player’s choice of n different factors under consideration. We will develop rules to predict (under specific assumptions about how each of the factors x1,…, xn are valued) the results in the cases of (a) no social communication and (b) complete social communication. In doing so, we will be able to develop and prove mathematical theorems that will predict the strategies and choices utilized by players in these two extreme situations. We will then generalize the problem to incorporate a social network among the players. This network will connect some players (allowing them to see each others’ choices) but not all, so that each player has less than total information. We will then study how the structure of the social network influences the strategies utilized by the players. We will begin this by considering some simple networks and directly calculate the strategies adopted by the players. We will then attempt to generalize these results to state and prove mathematical theorems connecting the properties of the social network to the strategies predicted for the players.