Question

Consider our nine-region location game to be a model of competition between two political candidates. Fifty voters are located in each of the nine regions of the political spectrum; each voter will vote for the closest candidate. Voters in regions equidistant from the two candidates will split evenly between them. (a) Assume that each candidate wants to maximize the number of votes he or she receives. Are the candidates' preferences the same as those modeled in the basic location game? Discuss how this setting differs from the setting in which candidates want to maximize the probability of winning (with the rule that the candidate with the most votes wins the election and, in case of a tie, each candidate is awarded the office with probability \(1 / 2\) ). (b) Suppose, contrary to the basic location model, that voters are not uniformly distributed across the nine regions. Instead, whereas regions \(1,2,3,4,5,8\), and 9 each contains fifty voters, regions 6 and 7 each contains \(x\) voters. Analyze the candidate location game for the case in which \(x=75\). In a rationalizable outcome, can one of the candidates win with probability 1 ? (c) What happens if \(x>75\) or \(x<75\) ?

Short answer

For (a), preferences align with vote maximization but differ in outcome assessment. For (b), one candidate can potentially win outright by focusing on regions 6 and 7. For (c), if x>75, the focus intensifies on regions 6 and 7; if x<75, distribution focus shifts.

Step-by-step solution

Understanding the Basic Location Game Model

In the basic model, two political candidates position themselves on a line with nine regions. Each candidate wants to capture the maximum voters by being closest to them on this line. Voters will choose the candidate who is nearest to them. If two candidates are equidistant from a group of voters, those voters are split evenly between the candidates.

Candidate Preferences in Vote Maximization - Part (a)

In this scenario, candidates seek to maximize their number of votes, similar to the location game where businesses maximize customers. However, the difference comes in the outcome evaluation: in the political model, a candidate wins outright if they have more votes, whereas in a tie, they have an equal chance of winning. Thus, while the structure of decision-making is similar, the evaluation differs because there's no direct advantage to splitting equidistant votes for winning probability purposes.

Analyzing Voter Distribution with x=75 - Part (b)

Given new voter distributions, regions 6 and 7 hold 75 voters each, while other regions hold 50 each. Candidates will strategize placements based on voter density to maximize votes. When both those regions hold a higher number of voters, candidates place themselves closest to regions 6 and 7 to capture a larger share of votes. If one candidate places optimally in this more dense region over the other, that candidate can win outright, maximizing votes and securing a decisive advantage.

Effects of x>75 or x

When \(x > 75\), regions 6 and 7 become even more attractive because capturing them can secure a majority. Each candidate will prioritize those regions even more to gain a decisive vote share. If \(x < 75\), while still relevant, other regions become comparatively more valuable. In either case, candidates must rebalance their positions to ensure they optimize voter coverage according to new distributions.

Key concepts

Political Strategy

In game theory, political strategy refers to the deliberate actions and decisions that candidates take to maximize their chance of winning an election. This involves strategic positioning, where candidates decide where to stand on issues that matter to voters. When two political candidates compete, they must consider various factors such as the distribution of voters, how to appeal to different groups, and whether to adjust their positions based on voter behavior.

For instance, in the location game model, candidates position themselves on a spectrum to get as close as possible to the largest number of voters. It's a balancing act: they must attract enough support without alienating potential swing voters. Political strategy isn't just about getting votes; it's about maximizing them effectively over various regions. Unlike in business, where maximizing customers might mean more profit, here the focus is on securing a win by adjusting strategies according to voter distribution and behavioral cues.

Voter Distribution

Voter distribution is a crucial factor in determining political strategies. In the exercise example, each region initially has the same number of voters except for regions 6 and 7, which are the focus due to alterations in distribution. This changes how candidates strategize because certain areas become more influential.

When adjusting voter distribution, say by increasing or decreasing the number of voters in a particular region, candidates might either focus on heavily populated areas to secure a win (when voter numbers are high) or shift focus to less populated regions (when those become pivotal). Understanding voter distribution allows candidates to effectively decide where they should focus their campaign efforts, often leading to highly targeted campaigns in specific regions.

Rationalizable Outcome

A rationalizable outcome in game theory is where both players make decisions based on strategic reasoning and anticipation of the other player's actions. In the context of the location game model, candidates must rationalize their positions based on where they expect their opponent to position themselves.

In the example scenario, if candidates see that regions 6 and 7 are crucial due to a higher concentration of voters (with 75 voters each), they will position themselves rationally in these areas. A rationalizable outcome ensures that each candidate acts logically, trying to optimize their vote share given any changes in voter distribution or competitor strategies. Thus, candidates continuously assess and adjust, leading to outcomes where each believes they cannot unilaterally improve their position based on possible moves of the opponent.

Location Game Model

The location game model is a framework in game theory that helps illustrate how competitors (in this case, political candidates) position themselves along a continuum to attract "customers" (or votes). It is analogous to businesses situating themselves in the best possible location to capture maximum market share. In a political context, candidates use this model to position themselves ideologically on issues to maximize vote acquisition.

In this nine-region scenario, candidates strategically decide where to "stand" on this spectrum. The key is understanding that voters select candidates nearest to them ideologically, similar to how consumers might choose the closest store. The candidates thus aim to cover the most significant swath of voter-rich territories, especially when facing uneven voter distribution across regions. This model helps candidates understand the importance of taking strategic positions based on voter proximity and numbers, thereby facilitating better decision-making in the electoral landscape.