## Abstract

We extend previous work on the role of politically motivated donors who contribute to candidates in an election with single dimension policy preferences. In a two-stage game wherein donors observe candidate policy positions and then allocate funding accordingly, we find that reducing the cost of donations incentivizes candidates to position closer to one another, reducing policy divergence. Furthermore, we find that as donations become more effective at influencing voter decisions, candidates respond less to voter preferences and more to those of donors. In addition, we analyze the presence of asymmetries in the model using numerical analysis techniques. We also extend our model by allowing for public funding from governments. By implementing stringent campaign contribution limits, candidate positions align with voter preferences at the cost of wider policy divergence. In contrast, unlimited campaign contributions lead to candidate positions moving away from voters to donors’ preferences, but increase policy convergence.

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## Notes

- 1.
Campaign spending levels in US presidential elections increased by 45% ($687 million to $1 billion) from 2004 to 2008, and again by 40% ($1 billion to $1.409 billion) from 2008 to 2012. In the UK’s parliamentary elections, in contrast, donations declined 32% ($36.5 million to $24.9 million) from 2005 to 2010 and subsequently increased by 12% ($24.9 million to $28 million) from 2010 to 2015.

- 2.
For an example of an election with three candidates, see Evrenk and Kha (2011).

- 3.
Additional work by Barro (1973) examines how candidates’ motivations may not coincide with those of their electoral base.

- 4.
Those results were extended in Calvert (1985), who demonstrated the assumptions under which policy convergence occurs. Empirical work by Morton (1993) tests the predictions and suggests that policy divergence occurs, but to a lesser extent than the theoretical prediction. Work by Zakharov and Sorokin (2014) suggests that the results also hold for a wide array of voting probability functions.

- 5.
Candidates’ announcements of policy positions might not be credible to donors or voters. Work by Alesina (1988) and Aragonès et al. (2007), however, suggest that when candidates face repeated elections, voters (and, by extension, donors) recall when implemented and announced policy positions differ, and form their beliefs on a candidate’s true position accordingly. As a result, candidates announce the policy they intend to implement and the announcements can be considered credible. We assume that candidates cannot shirk on their announced positions, but our setting can be extended to models wherein candidates build their reputation.

- 6.
Work by Hinich and Munger (1994) explores campaign contributions as a hindrance to a rival rather than a benefit to a preferred candidate.

- 7.
Empirical work estimates the effects of campaign contribution limits. Jacobson (1978) and Coate (2004) focus on how contribution limits influenced the margins of victory between candidates (namely, an incumbent against a challenger). Our model, however, examines the effect of contribution limits on equilibrium positioning, rather than on the margin of victory.

- 8.
In 2012, Mitt Romney faced a long primary challenge, which required him to spend 87% of the $153 million raised through June 2012, when he clinched the Republican nomination. In contrast, incumbent President Barack Obama spent 69% of the $303 million that he had raised over the same period. To generate additional funds, Romney courted donors who had either supported his rivals in the primary election or stayed out completely. Those donors were able to observe Romney’s policy positions long before ever offering him their aid.

- 9.
- 10.
Intuitively, both donors benefit from the contribution made to the supported candidate, and a situation similar to a public good arises, wherein donors free ride on each others’ contributions. In particular, every dollar donated to candidate

*i*by donor*l*reduces donor*k*’s contribution by exactly one dollar. - 11.
Intuitively, contributions from one donor are detrimental to the other donor’s preferred election outcome, and the other donor has incentive to contribute further to his own candidate to protect his interests in the election.

- 12.
Our model assumes a linear combination of the Downsian and Wittman utilities. While that assumption restricts the preferences of the candidates, the convexity of each candidate’s preferences is preserved. The authors wish to thank an anonymous referee for bringing that point to our attention.

- 13.
Many studies consider functions such as \(v_{i}(x_{i})=-A(x_{i}-{\hat{x}}_{i})^{2}\), where \(A>0\), which is negative everywhere except at its max when \(x_{i}={\hat{x}}_{i}\), i.e., the implemented policy coincides with candidate

*i*’s ideal, where \(v_{i}({\hat{x}}_{i})=0\). - 14.
In the first term on the right-hand side, an increase in \(x_{i}\) corresponds to candidate

*i*moving closer to (away from) the median voter, thus making his policy position more (less) attractive to voters and increasing (decreasing) the probability that he wins the election. The second term depicts how candidate*i*’s policy position affects the contributions he receives from each donor. We know that \(\frac{\partial p_{i}}{\partial D_{i}}>0\) since an increase in donations to candidate*i*increases his chances of winning the election. The signs of \(\frac{dk_{i}^{*}}{dx_{i}}\) and \(\frac{dl_{i}^{*}}{dx_{i}}\) depend on candidate*i*’s policy position relative to the ideal policies of donors*k*and*l*, respectively. - 15.
Since candidate

*i*was relatively close to donor*k*’s ideal policy position, and donors’ utility is concave by definition, candidate*i*’s shift in \(x_{i}\) causes only a small reduction in donor*k*’s expected utility, but yields to donor*l*a much larger gain in his expected utility. - 16.
Utilizing \(\alpha\) and \(\beta\), we can simulate several different population distributions. When \(\alpha =\beta >1\), we obtain a symmetric population of voters whose mean is 0.5 and are more concentrated towards the center of the distribution. Alternatively, when \(\alpha =\beta <1\), we obtain a symmetric population of voters whose mean is 0.5 but are more concentrated in the tails of the distribution. With \(\alpha>\beta >1\), we obtain a distribution that has a mean above 0.5 and has a negative skew. Lastly, with \(\beta>\alpha >1\), we obtain a distribution that has a mean below 0.5 and has a negative skew.

- 17.
The setup is a significant departure from Ball’s (1999a) original model, which did not consider the effectiveness of donations as a linear combination, but rather as a parameter (Ball refers to it as \(\gamma\)). We choose (5) as it guarantees that the probability that candidate

*i*wins the election falls strictly between 0 and 1. - 18.
The specification can lead to convexity issues. While we are unable to prove that the set of all Downsian and Wittman utilities are convex analytically, we examined several instances within those sets numerically. For the present analysis, all donation levels and the position of candidate

*j*, \(x_{j}\), were fixed, and only the values of candidate*i*’s policy position, \(x_{i},\) and the linear combination between the Downsian and Wittman specifications, \(\gamma\), were varied. We then calculated the resulting payoffs for candidate*i*. Generally, we find that candidate*i*’s best response function exhibits discontinuities only when \(\gamma\) is extremely small, as described later in this section, or in the special case in which both candidates*i*and*j*have the same ideal policy position. In those cases, we can show numerically that our model predicts policy convergence. Otherwise, candidate*i*’s best response function is continuous and the set of Downsian and Wittman utilities is convex around candidate*i*’s best response. - 19.
To approximate continuity, the numerical analysis also was performed using 2,001 and 5,001 equally spaced points, confirming that the results are unaffected. We then reproduced our simulations by drawing 1,001 points randomly on the [0, 1] interval, showing that the results essentially were identical to those using equally spaced points.

- 20.
For instance, if the parameters are \(\gamma =1\), \(\lambda =0.5\), \(w=0\), \(\alpha =\beta =2\), \({\hat{d}}_{k}=0.2\), \({\hat{d}}_{l}=0.8\), \({\hat{x}}_{i}=0.3\), \({\hat{x}}_{j}=0.7\), \({\bar{k}}={\bar{l}}=1\), \(\eta =0.5\) and \(c=0.03\), if we start with \(x_{j}=0.1\), candidate

*i*’s largest payoff occurs at \(x_{i}=0.3\), where his payoff becomes \(-0.0053\). - 21.
In that situation, candidate

*i*positions between his own ideal policy, \(\hat{x_{i}},\) and the location of the median voter, but prefers positioning closer to \(\hat{x_{i}}\). - 22.
- 23.
Intuitively, with symmetry, donor preferences also align with voter preferences.

- 24.
The asymmetry measurement is derived from the difference between each position and that of the median voter,

*m*, \(\left| x_{k}-m-(x_{l}-m)\right|\) which simplifies to our expression. Notably, when policy positions are equidistant from the median voter, the expression evaluates to zero. - 25.
When \(\gamma =0\), we return to policy convergence, as candidate policy preferences do not impact the Downsian equilibrium.

- 26.
That is the only case of complete policy convergence in the Wittman specification we found.

- 27.
The candidate receiving donations is more likely to win the election under such circumstances. The candidate who positions closer to the median voter (and receives no donations), however, still has a positive probability of winning. That scenario is reminiscent of the 1896 US presidential election won by William McKinley, an industrialist with strong backing by business interests. His opponent, William Jennings Bryan, adopted policy positions that were popular among the mass of voters, but was unable to raise money from potential donors. McKinley raised $3.5 million to Bryan’s $0.5 million, which led to McKinley winning the election.

- 28.
For calculated values of \(k_{1}\) and \(k_{2}\), refer to the selected simulation results table in “Appendix 3”.

- 29.
Which, as described in the numerical analysis section, produces more policy divergence than in the original Wittman (1983) model, but the candidates do not display the skewness towards donors’ ideal policies because \(\lambda >0\).

- 30.
Other countries imposing limits on campaign contributions are Uruguay, Belgium, Finland, France, Greece, Ireland, Poland, Japan and South Korea; with most limiting individual donations below $8000. Countries limiting political parity spending include Canada, Austria, Belgium, Czech Republic, France, Greece, Hungary, Ireland, Israel, Italy, Poland, Japan, New Zealand, and South Korea.

- 31.
Of note, intermediate values of \(\gamma\), the weight that every candidate places on policy implementation relative to the probability that they win the election, exist that yield policy convergence only for low values of \(\lambda\). For example, when \(\gamma =0.5\) with our assumed parameters, policy convergence occurs for \(\lambda <0.74\), but policies diverge slightly for values above that value.

- 32.
At these parts of the best response function, candidate

*i*obtains a higher probability of winning the election by distancing himself from his opponent and enabling donors to contribute to his own campaign (as donors will contribute approximately zero when candidates position next to one another). - 33.
Term \(N(D_{i}^{\eta }-D_{j}^{\eta })\) is our normally distributed contribution to the probability that candidate

*i*wins the election based on their received donations. - 34.
As described in Propositions 3B and 4B in Wittman’s (1983) paper. The net effect of an increase in \(\lambda\) is ambiguous, as it strongly depends on the symmetry of ideal policy positions, but in general, as \(\lambda\) increases, candidates have stronger incentives to deny donations to their opponent by moving closer to one another; as described in Corollary 4.

- 35.
We also find that for every cost \(c<{\bar{c}}\), no Nash equilibrium exists. Intuitively, as donations become extremely cheap, candidate behavior becomes erratic. Candidates receive large donations for even small deviations from their current positions, and constantly vie for the most donations from their respective donors. This causes no equilibrium to emerge. As a note, for large donations, the concavity property of our normal distribution also breaks down, which could be driving this result.

- 36.
In our numerical analysis,\(\gamma =1\), \(\lambda =0.5\), \(w=0\), \(\alpha =\beta =2\), \({\hat{d}}_{k}=0.2\), \({\hat{d}}_{l}=0.8\), \({\hat{x}}_{i}=0.3\), \({\hat{x}}\), \(\eta =0.5\), we obtain that for the value of \(c_{w}=0.053\), the Wittman model and the Wittman specification of our model yield the same equilibrium policy positions for both candidates.

- 37.
This includes extreme cases where candidates’ ideal policies are at the endpoints of the policy line, \({\hat{x}}_{i}=0\) and \({\hat{x}}_{j}=1\).

- 38.
Once again, this also holds for extreme ideal policies, \({\hat{x}}_{k}=0\) and \({\hat{x}}_{l}=1\).

- 39.
From disclosure website opensecrets.org, data obtained suggests that no major super PAC supports multiple candidates in the same election. This holds true for several elections, dating back to before 2008.

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We thank the Associate Editor, Keith Dougherty, and two anonymous referees for their helpful comments and suggestions. We are also grateful to Ron Mittelhammer, Raymond Batina, Ana Espinola-Arredondo and all seminar participants at Washington State University.

## Appendix

### Appendix

### Appendix 1: Further details on the numerical simulation

Downs (1957) specification (\(\gamma =0\)). Under the Downs (1957) specification, candidates seek to solely maximize their probability of winning the election. By setting both \(\gamma =0\) and \(\lambda =0\), we obtain the original model and result proposed by Downs in that each candidate maximizes his probability of winning the election by positioning himself at exactly the median voter. For values of \(\lambda >0\), campaign contributions also determine the probability of winning the election, and our results are presented in Fig. 6 below.

Figure 6a plots the results in the original Downs (1957) model since \(\lambda =0\). The best response for either candidate *i* is to position himself \(\varepsilon >0\) closer towards the median voter relative to his opponent. This behavior continues until both candidates converge at the median voter and have even odds of winning the election. In Fig. 6b, we have the Downsian specification of our model where \(\lambda >0\). Similar to the Downsian model, our model has every candidate *i* position himself closer to the median voter’s ideal position than his opponent, where the best response is to position \(\varepsilon >0\) closer to the median voter for positions near the median voter. As candidate *i*’s opponent deviates significantly from the median voter’s ideal policy position, however, candidate *i*’s best response is now to increase the distance between his own position and that of his opponent (flatter best response function when \(x_{j}\) is either high or low). Intuitively, at the more extreme points of the best response function, the ability to increase his probability of winning by targeting voter preferences diminishes quickly due to the concavity of the probability function.^{Footnote 32}

The equilibrium results of the original Downsian model and the Downsian specification of our model remain the same. Thus, the only Nash equilibrium we find is \(x_{i}^{*}=x_{j}^{*}=m\), the location of the median voter, and \(k_{i}^{*}=k_{j}^{*}=l_{i}^{*}=l_{j}^{*}=0\), as no donor has any incentive to donate to either candidate. Intuitively, in the Downsian specification of our model, as candidates approach the median voter donations to each candidate effectively disappear. Every candidate has incentive to position at the median voter to maximize his vote share, as well as incentive to deny campaign contributions to his opponent, as described in Corollary 4. Furthermore, lowering *c*, the marginal cost of donations only exacerbates this effect.

Wittman (1983) specification (\(\gamma =1\)). In the Wittman (1983) specification, instead of maximizing the probability of winning the election, every candidate *i* maximizes his expected policy outcome. As a result, candidates position themselves closer to their ideal policy position rather than the median voter’s ideal (as in the Downs (1957) model). When we set \(\gamma =1\) and \(\lambda =0\), we can obtain both the original model and results as presented by Wittman. Allowing for donations to affect the probability of winning the election (\(\lambda >0\)), we obtain the results in Fig. 7.

In Fig. 7a, we have the original Wittman (1983) model where \(\lambda =0\). Every candidate *i* positions himself close to his ideal policy position, i.e., the best response function \(x_{i}(x_{j})\) lies close to \({\hat{x}}_{i}\). As candidate *j* positions himself closer to the median voter, candidate *i* responds by moving rightward, but at a much slower rate than seen in the Downs (1957) specification (flatter best response function). This leads to an equilibrium where candidate policy positions diverge from the median voter and from their own ideal policies. Figure 7b contains the Wittman specification of our model where \(\lambda >0\). The key difference between the two models happens when \(x_{j}\) is relatively high. In this case, the best response of candidate *i* is to remain even closer to his own ideal policy position since candidate *j* positions himself significantly to the right of the median voter; see flat segment in the right-hand side of \(x_{i}(x_{j})\). The intuition is similar to that of the Downsian specification, as candidate *i* receives a larger benefit from campaign contributions rather than targeting voter preferences when his opponent positions himself at extreme locations.

The location of our equilibrium under the Wittman (1983) specification with donations can vary relative to the equilibrium of Wittman’s model, itself. Intuitively, by introducing donations we both lower voter sensitivity, and introduce bias for whichever candidate receives more donations, as described in Wittman’s (1983) paper. We can reproduce Wittman’s results by setting an inverse relationship between Wittman’s voter sensitivity parameter *s* and our donation effectiveness parameter, \(\lambda\); and a proportional relationship between Wittman’s bias parameter *B* and a combination of our \(\lambda\) and \(N(D_{i}^{\eta }-D_{j}^{\eta })\) terms.^{Footnote 33} As \(\lambda\) increases, voter sensitivity decreases, which causes candidates to move closer to their own ideal policy positions; and the bias increases in magnitude, which causes candidates to move towards whichever candidate the bias favors.^{Footnote 34}

As the marginal cost of donations, *c*, decreases, donors contribute more to their preferred candidates. This influx of donations causes both candidates to position closer to each other, since each candidate is able to deny donations to his opponent; as explained in Corollary 4. Due to the concavity of the donors’ utility functions, having a candidate that is positioned farther away from the donor move closer entails a much larger increase in utility than having a candidate that is close to the donor move away. Thus, candidate *i* can significantly reduce the donations that candidate *j* receives by moving slightly closer to candidate *j*’s policy position while only experiencing a slight decrease in the amount of donations that he receives.^{Footnote 35}

In summary, when the marginal cost of donations, *c*, is sufficiently high \(c>c_{w}\) (low \(c<c_{w}\)), our results predict less (more) convergence than in Wittman’s model.^{Footnote 36} When \(\lambda =0\) (as in the original Wittman model), candidates position between their own ideal policy positions and the ideal policy position of the median voter. As \(\lambda\) increases, candidates put less weight on the location of the median voter’s ideal policy and more weight on the source of the bias, the donors’ ideal policy position. At the extreme case of \(\lambda =1\), candidates disregard the median voter’s ideal policy position entirely when they receive donations, and position themselves between their own ideal position and the midpoint of the donors’ ideal policies, \(\frac{{\hat{d}}_{k}+{\hat{d}}_{l}}{2}\).

Other effects are similar to those described in Wittman (1983). If the ideal policy position of either candidate moves rightward (leftward), both candidates move rightward (leftward) in equilibrium.^{Footnote 37} Intuitively, if candidate *j*’s ideal policy position moves rightward, he positions closer to it, which induces candidate *i* to also move rightward to receive more votes and to deny candidate *j* additional donations. Likewise, if either donor moves his most preferred policy position rightward (leftward), both candidates move their equilibrium policy position rightward (leftward).^{Footnote 38}

Mixed specification (\(0<\gamma <1\)). Using a mixed specification, we obtain results that fall between the Downs (1957) and Wittman (1983) specifications. When \(\gamma\) is low (\(\gamma <0.655\) in our example), we find that policy convergence at the median voter occurs. For values of \(\gamma\) above this threshold, policy positions diverge until they reach those at the Wittman (1983) specification when \(\gamma =1\). Interestingly, the best response functions for both candidates show properties of both the Downsian and Wittman models as shown in Fig. 8. Their behavior depends on each candidate’s location relative to their ideal position.

Without loss of generality, when candidate *i* prefers a leftward policy position than the median voter and candidate*j* prefers a rightward policy position than the median voter, i.e., \({\hat{x}}_{i}<m<{\hat{x}}_{j}\), for low values of \(x_{j}\), candidate *i*’s best response is to target the voters consistent with the Downsian model as seen in Fig. 8a, and he positions himself \(\varepsilon >0\) closer to the median voter than candidate *j*. This occurs up until a policy point above the median voter, where candidate *i* no longer moves his position rightward in response to a rightward move in candidate *j* ’s position. At this point, candidate *i*’s expected policy payoff dominates his preference to maximize his probability of winning the election, and he behaves more in line with the Wittman model. An analogous argument applies for candidate *j*’s best response to candidate *i*’s position. In this case, both candidates position themselves at exactly the median voter in equilibrium, and policy convergence occurs. In equilibrium, neither donor contributes to either candidate’s campaign, and the election is decided by a coin flip.

In Fig. 8b, we have the case where \(\gamma\) is large enough to induce policy diversion. The major difference in this case is that both candidates shift from maximizing their probability of winning the election to maximizing their expected policy payoff at a position below (above for candidate *j*) the median voter. This leads to behavior more in line with the Wittman specification rather than the Downsian, where a single Nash equilibrium in pure strategies exists where candidates select different policy positions, ones that are closer to their most ideal position.

### Appendix 2: Comparing candidate positions against donor’s ideals

Figure 9a plots candidate equilibrium position as a function of the marginal cost of donations, *c*. Starting from the right side of the figure, when *c* is large, every candidate receives fewer donations and has little incentive to position closer to his opponent in order to deny him of those donations. As a result, policy divergence is higher with large values of *c*. As *c* decreases, each candidate positions closer to his rival, as he has strong incentives to deny his rival of the additional donations that are available due to the reduced marginal cost.

Figure 9b depicts candidate equilibrium position as a function of the effectiveness of donations, \(\lambda\). In this figure, candidate *i* has an ideal policy position closer to the midpoint between the donors’ ideal policy position (donors are slightly asymmetric towards candidate *i*), and thus, candidate *i* responds quickly to an increase in \(\lambda\) by moving closer to his own ideal policy position. Candidate *j* follows at a slower rate, increasing policy divergence for low values of \(\lambda\). As \(\lambda\) increases further, candidate *i* responds less to further increases (his line becomes flatter), and candidate *j* is able to deny more donations to his rival. For high values of \(\lambda\), we observe decreased policy divergence.

### Appendix 3: Candidate positions with donation constraints

For parameter values of \({\hat{d}}_{k}=0.2\), \({\hat{d}}_{l}=0.8\), \({\hat{x}}\), \({\hat{x}}_{j}=0.7\), \(\eta =0.5\), \(\lambda =0.5\), and \(a=\beta =2\), the following results were obtained:

\({\bar{k}}={\bar{l}}\) | Equilibrium Values \((x_{i}^{*},x_{j}^{*})\) | ||
---|---|---|---|

\(c=0.01\) | \(c=0.02\) | \(c=0.03\) | |

0.8 | (0.346, 0.654) | (0.346, 0.654) | (0.346, 0.654) |

0.9 | (0.346, 0.654) | (0.346, 0.654) | − |

1.2 | (0.346, 0.654) | (0.346, 0.654) | − |

1.3 | (0.346, 0.654) | (0.346, 0.654) | (0.387, 0.613) |

1.5 | (0.346, 0.654) | (0.346, 0.654) | (0.387, 0.613) |

1.6 | (0.346, 0.654) | − | (0.387, 0.613) |

2.1 | (0.346, 0.654) | − | (0.387, 0.613) |

2.2 | − | − | (0.387, 0.613) |

2.5 | − | − | (0.387, 0.613) |

2.6 | − | (0.398, 0.602) | (0.387, 0.613) |

4.1 | − | (0.398, 0.602) | (0.387, 0.613) |

4.2 | (0.422, 0.578) | (0.398, 0.602) | (0.387, 0.613) |

10 | (0.422, 0.578) | (0.398, 0.602) | (0.387, 0.613) |

For \(c=0.01\), the low marginal cost of donation allows donors to contribute large donations to their respective candidates. When donation constraints are set low, \({\bar{k}}<k_{1}=2.1\) in this case, candidates behave as if no donations are received and position themselves at \((x_{i}^{*},x_{j}^{*})=(0.346,0.654)\). For values of \(k_{1}<{\bar{k}}<k_{2}=4.2\), no equilibrium exists, as candidates leverage constraints on their opponents to position closer to their own ideal policy positions. Lastly, when \({\bar{k}}\), candidates act as if they were unconstrained and position at \((x_{i}^{*},x_{j}^{*})=(0.422,0.578)\).

As we increase *c* to 0.02 or 0.03, we find that the values of \(k_{1}\) and \(k_{2}\) decrease. For example, when \(c=0.02\), the higher marginal cost of donations causes donors to reduce their contribution levels, and thus the breakpoints for each scenario must also decrease. We find that \(k_{1}=1.5\) and \(k_{2}=2.6\) when \(c=0.02\); and \(k_{1}=0.9\) and \(k_{2}=1.3\) when \(c=0.03\). Of note, the fully constrained equilibrium does not change when\({\bar{k}}<k_{1}\) regardless of the value of *c* since candidates behave as if they receive no contributions, but as *c* decreases, the unconstrained equilibria show reduced policy divergence.

### Appendix 4: Public funding numerical results

The presence of public funding in our model behaves qualitatively similar to adding an asymmetry. For example, when \(\gamma\) is low (as in the Downsian Specification), equal public funding levels among candidates retains our equilibrium at the median voter. For even small public funding donation advantages, however, we arrive at situations where no equilibrium in pure strategies exists, much like the cases described in the asymmetry section.

In contrast, when \(\gamma\) is high (as in the Wittman Specification), a public funding advantage does not prevent the emergence of an equilibrium in pure strategies. Under these conditions, both candidates again behave as if an asymmetry were present, positioning closer to the candidate with the public funding advantage, the results of which are shown below in Fig. 10.

Figure 10a depicts candidate equilibrium positions as a function of \(F_{i}\) while \(F_{j}\) is held constant at 0.5 and \(\lambda\) is held constant at 0.5. For comparison purposes, we denote point \({\bar{x}}_{i}\) as candidate *i*’s equilibrium policy position without public funding. As seen in the figure, for values of \(F_{i}<0.5\), equilibrium positions are skewed rightward, towards candidate *j*’s ideal. As \(F_{i}\) increases, however, the skewness at first disappears at \(F_{i}=F_{j}=0.5\), and then becomes skewed leftward as \(F_{i}\) increases further above 0.5. Figure 10b depicts candidate equilibrium positions as a function of \(\lambda\) with \(F_{1}=0.8\) and \(F_{2}=0.5\). In this situation, we again observe the increased policy convergence as \(\lambda\) increases, as seen in Fig. 4b (as an increase in \(\lambda\) increases the effect of private, as well as public donations). However, we do observe an asymmetry in favor of candidate *i*, due to their advantage in public funding. As \(\lambda\) increases, candidates shift their priorities from the voters to the donors, but candidate *i*’s public donation advantage also shifts both candidates more towards candidate *i*’s ideal policy.

### Proof of Proposition 1

###
**Lemma 1**

*Donor*
*k*
*’s equilibrium donation to candidate*
*i*
*,*
\(k_{i}\)
*, solves*

Intuitively, donor *k* increases his contribution \(k_{i}\) until his marginal benefit from further donations (left-hand side of Eq. (6)) coincides with his marginal cost (right-hand side of (6)). Note that the marginal benefit captures the additional probability that candidate *i* wins the election thanks to larger donation, \(\frac{dp_{i}}{dD_{i}}\ge 0\), and the utility gain that donor *k* obtains when candidate *i* wins the election to candidate *j*, \(u_{k}(x_{i};{\hat{d}}_{k})-u_{k}(x_{j};{\hat{d}}_{k})\). Needless to say, if donor *k* prefers candidate *j* winning the election then \(u_{k}(x_{i};{\hat{d}}_{k})<u_{k}(x_{j};{\hat{d}}_{k})\), and the left-hand side becomes unambiguously negative, ultimately yielding a corner solution where donor *k* does not contribute to candidate *i*’s campaign in equilibrium. This result suggests that ever donor *k* will only contribute to the candidate yielding the highest utility; as we prove in the next lemma.

###
**Lemma 2**

*In equilibrium, every donor*
*k*
*contributes to one candidate at most.*

###
*Proof*

Assume that donor *k* contributes to both candidates *A* and *B*, i.e., \(k_{i},k_{j}>0\) and thus Eq. (6) binds with equality for candidates *i* and *j*, i.e.,

Setting Eqs. (7) and (8) equal to one another and simplifying yields

which cannot hold, since the left side of Eq. (9) is positive, while the right side is negative. Therefore, donor *k*’s contribution to at least one of the candidates must equal zero. \(\square\)

Intuitively, if donor *k* contributes to candidate *i*, he does so to increase the probability that candidate *i* wins the election. On the contrary, any contribution that donor *k* makes to the other candidate \(j\ne i\) lowers the probability that candidate *i* wins the election. Thus, contributions to both candidates are counterproductive, and every donor *k* only donates to the candidate whose policy position yields him the highest utility level.^{Footnote 39} As a remark, note that if both policy positions yield the same utility for donor *k*, \(u_{k}(x_{i};{\hat{d}}_{k})=u_{k}(x_{j};{\hat{d}}_{k})\) then his marginal benefit of contributing to candidate *i* (left-hand side of (2)) becomes nil, inducing no donations to either candidate, i.e., \(k_{i}^{*}=0\) for all *i*.

Lemmas 1 and 2 allow us to characterize the solution of the second stage of the game into several cases, as detailed in Proposition 1. \(\square\)

### Proof of Corollary 1

First, we show that \(\frac{dk_{i}^{*}}{dl_{i}}=-1\). Differentiating Eq. (2) with respect to \(l_{i}\) yields

where the only value that can satisfy the above equation is \(\frac{dk_{i}^{*}}{dl_{i}}=-1\).

Next, we show that \(\frac{dk_{i}}{dl_{j}}>0\). Using equation (2) with respect to \(k_{i}^{*}\) and \(l_{j}^{*}\), we have

Setting these two equations equal to one another and rearranging terms yields

Using the implicit function theorem,

0where the signs of \(u_{k}(x_{i})-u_{k}(x_{j})\) and \(u_{l}(x_{i})-u_{l}(x_{j})\) are by definition and the signs of \(\frac{d^{2}p}{dD_{i}^{2}}\) and \(\frac{d^{2}p}{dD_{j}^{2}}\) are due to the concavity and convexity, respectively of \(D_{i}\) and \(D_{j}\) on *p*. \(\square\)

### Proof of Corollary 2

Differentiating Eq. (2) with respect to \(x_{i}\) and \(x_{j}\) yields

Since the probability that candidate *i* wins the election is increasing and concave, we know that \(\frac{dp}{dD_{i}}>0\) and \(\frac{d^{2}p}{dD_{i}^{2}}<0\). Likewise, if donor *k* is contributing to candidate *i*, \(u_{k}(x_{i})-u_{k}(x_{j})>0\). The sign of \(\frac{du_{k}(x_{i})}{dx_{i}}\)\(\left( \frac{du_{k}(x_{i})}{dx_{j}}\right)\) can be determined by candidate *i*’s (*j*’s) position relative to donor *k*’s ideal position. If \(x_{i}<x_{k}\) (\(x_{j}<x_{k}\)), a rightward move in candidate *i*’s (*j*’s) policy position will entail an increase in the utility that donor *k* receives from that position, and thus \(\frac{du_{k}(x_{i})}{dx_{i}}>0\)\(\left( \frac{du_{k}(x_{i})}{dx_{j}}>0\right)\). On the contrary, if \(x_{i}>x_{k}\) (\(x_{j}>x_{k}\)), a rightward move in candidate *i*’s (*j*’s) policy position will entail an decrease in the utility that donor *k* receives from that position, and thus \(\frac{du_{k}(x_{i})}{dx_{i}}<0\)\(\left( \frac{du_{k}(x_{i})}{dx_{j}}<0\right)\). This leaves \(\frac{dk_{i}^{*}}{dx_{i}}\)\(\left( \frac{dk_{i}^{*}}{dx_{j}}\right)\) as the only unknown sign in the above equation. In order for the equation to hold with equality, it is necessary that \(\frac{dk_{i}^{*}}{dx_{i}}\)\(\left( \frac{dk_{i}^{*}}{dx_{j}}\right)\) have the same (opposite) sign as \(\frac{du_{k}(x_{i})}{dx_{i}}\)\(\left( \frac{du_{k}(x_{i})}{dx_{j}}\right)\), i.e., \(\frac{dk_{i}^{*}}{dx_{i}}>0\)\(\left( \frac{dk_{i}^{*}}{dx_{j}}<0\right)\) if \(x_{i}<x_{k}\) (\(x_{j}<x_{k}\)) and \(\frac{dk_{i}^{*}}{dx_{i}}<0\)\(\left( \frac{dk_{i}^{*}}{dx_{j}}>0\right)\) if \(x_{i}>x_{k}\) (\(x_{j}>x_{k}\), respectively).\(\square\)

### Proof of Lemma 2

Equilibrium location pairs \((x_{i}^{*},x_{j}^{*})\) must satisfy Eq. (4) for both candidates, which will depend on the signs of each term in those equations. Unambiguously, we know that \(\frac{dp_{i}}{dD_{i}}>0\), since by assumption, if candidate *i* receives more donations from either candidate, their subjective probability of winning the election will increase. Likewise, we have \(\frac{dp_{i}}{dD_{j}}<0\), as an increase in the amount of donations received by candidate \(j\ne i\) causes candidate *i*’s subjective probability to decrease. As shown in Corollary 1, when the subjective probability that candidate *i* wins the election is concave and \(x_{i}<\min \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\), we have that \(\frac{dk_{i}^{*}}{dx_{i}}>0\), \(\frac{dk_{i}^{*}}{dx_{j}}<0\), \(\frac{dl_{i}^{*}}{dx_{i}}>0\), and \(\frac{dl_{i}^{*}}{dx_{j}}<0\). In addition, since \(x_{i}<{\hat{x}}_{i}\), a rightward move in candidate *i*’s position increases the utility he receives if he wins the election, thus \(\frac{dv_{i}(x_{i})}{dx_{i}}>0\). Due to these relationships, the left-hand side of Eq. (4) is positive, and thus, it cannot hold with equality. Thus, any \(x_{i}<\min \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\) cannot be a solution to stage 1. A similar approach when \(x_{i}>\max \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\) shows that the left-hand side of Eq. (3) is unambiguously negative and also cannot solve Eq. (4).

When \(x_{i}=\min \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\) (\(x_{i}=\max \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\)), a similar situation occurs. All signs described in the previous paragraph are identical except for the sign that corresponds with \(\min \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\) (\(\max \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\)). This term is equal to zero, as candidate *i* is either receiving the most possible subjective probability contribution by positioning himself at the median voter, the most possible utility by positioning at his own ideal policy, or the most possible donations from the donor with the leftmost (rightmost) ideal policy position by positioning at his most preferred policy position. The outcome is the same where the left side of Eq. (4) is unambiguously positive (negative), and cannot be satisfied when \(x_{i}=\min \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\) (\(x_{i}=\max \{{\hat{x}}_{i},{\hat{x}}_{j},{\hat{d}}_{k},{\hat{d}}_{l},m\}\), respectively). \(\square\)

### Proof of Corollary 4

We can show the effect of rightward move of candidate *i*’s policy position, \(x_{i}\), graphically below in Fig. 11.

As seen in the above figure, when candidate *i* moves from position \(x_{i}\) to position \(x_{i}^{\prime }\) the decrease in utility to donor *k*, \(\Delta u_{k}(x_{i})\) is much less than the gain in utility to donor *l*, \(\Delta u_{l}(x_{i})\). Holding all other values of Eq. (2) constant, the resulting increase in candidate *i*’s policy position requires both equilibrium donation levels to decrease, but that for donor *l* to decrease by a larger amount. \(\square\)

### Proof of Lemma 3

Let \(x_{1}=f_{1}(x_{2})\) and \(x_{2}=f_{2}(x_{1})\) be two continuous functions of the argument \(x_{1}\) and \(x_{2}\) each having domain [*a*, *b*] where \(a,b\in [0,1]\) and range [0, 1]. An equilibrium is defined by solving the two equations simultaneously for the solution \(x_{1}^{*}\) and \(x_{2}^{*}\). The solution can be characterized via substitution as \(x_{1}^{*}=f_{1}\circ f_{2}(x_{1}^{*})=f(x_{1}^{*})\) and \(x_{2}^{*}=f_{2}(x_{1}^{*})\), where the composite function \(f=f_{1}\circ f_{2}\) is also continuous.

Define an equally-spaced grid on the domain consisting of \(n+1\) points, as

Define piecewise linear approximations to the composite function \(f=f_{1}\circ f_{2}\) as

where \(I_{A}(x)\) is an indicator function taking values \(I_{A}(x)=1\) if \(x\in A\) and \(I_{A}(x)=0\) if \(x\notin A\).

For compactness, define \(\Delta _{k,k+1}\equiv \frac{f(x_{k+1})-f(x_{k})}{x_{k+1}-x_{k}}\). Let \({\hat{x}}_{1}^{*}\) represent the linear piecewise-approximated equilibrium value that solves \({\hat{x}}_{1}^{*}={\hat{f}}({\hat{x}}_{1}^{*})\), i.e., a fixed point of the above piecewise linear approximation \({\hat{f}}(x)\). Then, either \({\hat{x}}_{1}^{*}=1\) if \({\hat{f}}(1)=f(1)=1\), or else \({\hat{x}}_{1}^{*}={\hat{f}}({\hat{x}}_{1}^{*})\) for all \({\hat{x}}_{1}^{*}\in [x_{k},x_{k+1})\). We can rewrite the last expression as \({\hat{x}}_{1}^{*}-{\hat{f}}({\hat{x}}_{1}^{*})=0\) which, by the definition of \({\hat{f}}(x)\), expands as follows

We now focus on term \(({\hat{x}}_{1}^{*}-x_{k})\Delta _{k,k+1}\) of expression (11). Note that, upon letting the number of equally spaced grid points in the [*a*, *b*] interval increase without bound, i.e., \(n\rightarrow \infty\), it follows that \(x_{k+1}\rightarrow x_{k}^{+}\), yielding

Using this result in expression (11), it follows that, in the limit as \(n\rightarrow \infty\),

and the approximate and exact equilibriums converge. \(\square\)

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### Cite this article

Dunaway, E., Munoz-Garcia, F. Campaign contributions and policy convergence: asymmetric agents and donations constraints.
*Public Choice* **184, **429–461 (2020). https://doi.org/10.1007/s11127-019-00732-1

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### Keywords

- Asymmetries
- Policy preferences
- Policy divergence
- Donors
- Political contributions
- Campaign limits

### JEL Classification

- C63
- C72
- D72