Campaign contributions and policy convergence: asymmetric agents and donations constraints

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.

Appendix

1.1 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.

Fig. 6

Downsian specification with $\lambda =0$ and $\lambda >0$, respectively

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.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.

Fig. 7

Wittman specification with $\lambda =0$ and $\lambda >0$, respectively

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.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.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.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.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.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).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.

Fig. 8

Mixed specification with ${\hat{x}}_{i}<m<{\hat{x}}_{j}$ for low and high values of $\gamma$

Without loss of generality, when candidate i prefers a leftward policy position than the median voter and candidatej 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.

1.2 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.

Fig. 9

Equilibrium candidate positions as a function of c and $\lambda$ when $\gamma =1$

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.

1.3 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.

1.4 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.

Fig. 10

Equilibrium candidate positions as a function of $F_{i}$ and $\lambda$ when $\gamma =1$

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.

1.5 Proof of Proposition 1

Lemma 1

Donor k ’s equilibrium donation to candidate i , $k_{i}$ , solves

$$\begin{aligned} \frac{dp_{i}}{dD_{i}}\left[ u_{k}(x_{i};{\hat{d}}_{k})-u_{k}(x_{j};{\hat{d}}_{k}) \right] \le c \end{aligned}$$

(6)

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.,

$$\begin{aligned} \frac{dp}{dD_{i}}\left[ u_{k}(x_{i})-u_{k}(x_{j})\right]= & {} 1 \end{aligned}$$

(7)

$$\begin{aligned} \frac{dp}{dD_{j}}\left[ u_{k}(x_{i})-u_{k}(x_{j})\right]= & {} 1 \end{aligned}$$

(8)

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

$$\begin{aligned} \frac{dp}{dD_{i}}=\frac{dp}{dD_{j}} \end{aligned}$$

(9)

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.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. $\square$

1.6 Proof of Corollary 1

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

$$\begin{aligned} \overbrace{\frac{d^{2}p}{dDi^{2}}}^{(-)}\left( \frac{dk_{i}^{*}}{dl_{i}} +1\right) \overbrace{\left[ u_{k}(x_{i})-u_{k}(x_{j})\right] }^{(+)}=0 \end{aligned}$$

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

$$\begin{aligned} \frac{dp}{dD_{i}}\left[ u_{k}(x_{i})-u_{k}(x_{j})\right]= & {} 1 \\ \frac{dp}{dD_{j}}\left[ u_{l}(x_{i})-u_{l}(x_{j})\right]= & {} 1 \end{aligned}$$

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

$$\begin{aligned} \frac{dp}{dD_{i}}\left[ u_{k}(x_{i})-u_{k}(x_{j})\right] -\frac{dp}{dD_{j}} \left[ u_{l}(x_{i})-u_{l}(x_{j})\right] =0 \end{aligned}$$

Using the implicit function theorem,

$$\begin{aligned} \frac{dk_{i}}{dl_{j}}=\frac{\overbrace{\frac{d^{2}p}{dD_{j}^{2}}}^{(+)} \overbrace{\left[ u_{l}(x_{i})-u_{l}(x_{j})\right] }^{(-)}}{\underbrace{ \frac{d^{2}p}{dD_{i}^{2}}}_{(-)}\underbrace{\left[ u_{k}(x_{i})-u_{k}(x_{j}) \right] }_{(+)}}>0 \end{aligned}$$

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. $\square$

1.7 Proof of Corollary 2

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

$$\begin{aligned} \overbrace{\frac{d^{2}p}{dDi^{2}}}^{(-)}\frac{dk_{i}^{*}}{dx_{i}} \overbrace{\left[ u_{k}(x_{i})-u_{k}(x_{j})\right] }^{(+)}+\overbrace{\frac{ dp}{dD_{i}}}^{(+)}\frac{du_{k}(x_{i})}{dx_{i}}= & {} 0 \end{aligned}$$

(10)

$$\begin{aligned} \underbrace{\frac{d^{2}p}{dDi^{2}}}_{(-)}\frac{dk_{i}^{*}}{dx_{j}} \underbrace{\left[ u_{k}(x_{i})-u_{k}(x_{j})\right] }_{(-)}-\underbrace{ \frac{dp}{dD_{i}}}_{(+)}\frac{du_{k}(x_{j})}{dx_{j}}= & {} 0 \end{aligned}$$

(11)

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).$\square$

1.8 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). $\square$

1.9 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.

Fig. 11

Donor utility functions

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. $\square$

1.10 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

$$\begin{aligned} {\mathcal {G}}=\left\{ a,a+\frac{b-a}{n},a+\frac{2(b-a)}{n},\ldots ,b\right\}\,{\text{for\,all}}\,i=1,2. \end{aligned}$$

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

$$\begin{aligned} {\hat{f}}(x)=\sum _{k=1}^{n}\left[ f(x_{k})+(x-x_{k})\frac{f(x_{k+1})-f(x_{k})}{ x_{k+1}-x_{k}}\right] I_{[x_{k},x_{k+1})}(x)+f(x_{n+1})I_{\{n+1\}}(x) \end{aligned}$$

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

$$\begin{aligned} {\hat{x}}_{1}^{*}-{\hat{f}}({\hat{x}}_{1}^{*})={\hat{x}}_{1}^{*}-\, \underset{{\hat{f}}({\hat{x}}_{1}^{*})}{\underbrace{\left( f(x_{k})+({\hat{x}} _{1}^{*}-x_{k})\Delta _{k,k+1}\right) }}\,=0\,{\text{ for\,all}}{\hat{x}}_{1}^{*}\in [x_{k},x_{k+1}). \end{aligned}$$

(12)

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

$$\begin{aligned}&\lim _{x_{k+1}\rightarrow x_{k}^{+}}\,({\hat{x}}_{1}^{*}-x_{k})\Delta _{k,k+1} \\&\quad =\lim _{x_{k+1}\rightarrow x_{k}^{+}}\,({\hat{x}}_{1}^{*}-x_{k})\times \lim _{x_{k+1}\rightarrow x_{k}^{+}}\,\frac{ f(x_{k+1})-f(x_{k})}{x_{k+1}-x_{k}} \\&\quad =0\times \lim _{x_{k+1}\rightarrow x_{k}^{+}}\,\frac{ f(x_{k+1})-f(x_{k})}{x_{k+1}-x_{k}}=0 \end{aligned}$$

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

$$\begin{aligned} {\hat{x}}_{1}^{*}-{\hat{f}}({\hat{x}}_{1}^{*})={\hat{x}}_{1}^{*}-\left[ f(x_{k})+0\right] =0 \end{aligned}$$

and the approximate and exact equilibriums converge. $\square$