Three-candidate election strategy

Abstract

The probability of a given candidate winning a future election is worked out in closed form as a function of (i) the current support rates for each candidate, (ii) the relative positioning of the candidates within the political spectrum, (iii) the time left to the election, and (iv) the rate at which noisy information is revealed to the electorate from now to the election day, when there are three or more candidates. It is shown, in particular, that the optimal strategy for controlling information can be intricate and non-trivial, in contrast to a two-candidate race. A surprising finding is that for a candidate taking the centre ground in an electoral competition among a polarized electorate, certain strategies are fatal in that the resulting winning probability for that candidate vanishes identically.

Keywords: electoral competition; measure change; signal processing; stochastic filtering.

Figure 1.

Winning likelihood. The probability that candidate 0 will win the election in 18 months (T = 1.5), as a function of the current support rate p for the candidate. The realized likelihood of winning a future election is always higher than today’s poll if p > 1/2; and conversely lower than the poll if p < 1/2. How much the winning probability deviates from the current poll depends on how much information is revealed over the next 18 months. Here, two examples are shown, corresponding to the values σ = 0.2 (in purple) and σ = 1.2 (in red).

Figure 2.

Winning probabilities as functions of (p₁, p₂). The probabilities of winning a future election to take place in 1 year's time (T = 1), when the information flow rate is set at σ = 1, are plotted here for the parameter choice (x₁, x₂, x₃) = (1, 2, 3). The forms of the probabilities for candidate 1 (a, in red) and candidate 3 (c, in blue) are entirely symmetric. However, the behaviour of the probability for candidate 2 (b, in purple) is slightly different in that there is a region in the parameter space (p₁, p₂) of the current support rates for which the probability of candidate 2 winning is identically zero. We will have more to say about this in the next section.

Figure 3.

Dynamical behaviours of the poll statistics {π_it} and the corresponding winning probabilities. Sample paths for the support rates (π_1t, π_2t, π_3t) for the three candidates are shown in (a,c). The corresponding winning probability processes for each candidate are shown in (b,d). The parameters are chosen as (x₁, x₂, x₃) = (1, 2, 3) for the values of the random variable X, (p₁, p₂, p₃) = (0.38, 0.26, 0.36) for the current support level so that the electorates are slightly polarized, and T = 1 year for the time left to the election day. Panels (a,b) correspond to the value σ = 0.25 for the information flow rate. In this case, the probability for the second candidate to win the election is identically zero. For a comparison, the corresponding results for the choice σ = 1 are plotted in (c,d), in which the second candidate narrowly secures a victory.

Figure 4.

Winning probabilities as functions of σ. The probability of winning an election in 1 year's time (T = 1), as a function of the information flow rate σ, is shown for the three candidates, labelled according to x₁ = 1, x₂ = 2 and x₃ = 3. The current poll statistics are taken to be p₁ = 0.38 for the first candidate on the left (red), p₂ = 0.26 for the second candidate taking the centre ground (purple), and p₃ = 0.36 for the third candidate on the right (blue).

Figure 5.

Gains in winning probabilities as functions of σ. If the political positioning (x₁, x₂, x₃) = (1, 2, 3) considered in figure 4 is shifted, how would that affect the winning probabilities? Here, the difference of the resulting winning probabilities to the one in figure 4 is shown for three different cases: (x₁, x₂, x₃) = (0.1, 2, 3.9) (a), (x₁, x₂, x₃) = (1, 2, 3.9) (b) and (x₁, x₂, x₃) = (1.5, 2, 3.9) (c). Other parameters are kept unchanged (p₁ = 0.38, p₂ = 0.26, p₃ = 0.36, and T = 1). If the difference is negative, then clearly the shift is disadvantageous. The result shows that among a polarized electorate, if the candidate on the left of the political spectrum leans further to the left and the candidate on the right leans further to the right, then this is generally disadvantageous for both. However, if the competition is dominated by noise (small σ values), then the candidate on the right can benefit by leaning further to the right.

Figure 6.

Maximum attainable support rates ${\pi }_{kT}({\xi }_k^{\ast })$ for the five candidates. For a range of values for the information flow rate σ, the maximum values of {π_kT} on the election day are shown by the dots, interpolated by lines to make the comparison easy. The current support rates {p_k} are given by the bottom values (in purple). In (a), the five candidates are all assumed to have an equal support rate of 20%, whereas they are chosen at random in (b,c). The results show how the maximum attainable support rates for different candidates vary rather dramatically, depending on the existence of candidates having different political leanings and their associated current support rates. The parameters are chosen to be (x₁, x₂, x₃, x₄, x₅) = (1, 2, 3, 4, 5) for the positioning of the candidates and T = 1 year for time left to the election.