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It is my first time here. In a dynamic model of electoral competition ($t=1,2,\ldots$), I want to prove incumbency advantage, i.e., that the incumbent is always re-elected (unless an exogenous event occurs). It all boils down to the following result, which I am confident is true (all simulations work), but I have not been able to prove yet. Let $c,\eta,\mu,i$ be parameters such that $0<c$, $1<\eta \leq 2$, and $0<i<\mu$. Define $r\in (i,\mu)$ and $l\in (-\mu,i)$ such that $$2(\mu-r)=c\eta (r-i)^{\eta-1}$$ and $$2(l+\mu)=c\eta (i-l)^{\eta-1}.$$ It is easy to see that $r$ and $l$ are uniquely defined within the given ranges. Then, I would like to prove that $$r^2+c(r-i)^{\eta} \leq l^2+c(i-l)^{\eta}.$$ In this problem, $i$ is the status-quo policy, $c$ and $\eta$ determine the cost of moving policies, $\mu$ is the degree of polarization of political parties, and $r$ ($l$) is the policy that a right-wing (left-wing) candidate would choose if elected. The inequality that I want to prove then simply says that the utility that the median voter will derive if $r$ is elected is at least as much as the one he/she will derive if $l$ is elected. We can assume that $r$ is the incumbent. If $\eta=2$, the result is straightforward. Together with my coauthors, I have already spent some time (in vain) trying to prove the claim. I have the impression that there has to be some easy argument, but in all proof strategies I have tried expressions turn extremely long and cumbersome immediately. By simulations, I also know that if $\eta>2$ or $i<0$ the result does not hold, so $1<\eta\leq2$ and $0<i<\mu$ should be used in the proof in some step. Any help or hint would be greatly appreciated. Thank you for your time!

My research question in a dynamic model of political competition boils down to the following conjecture. I am confident that it holds (all simulations work), but I have not been able to prove it yet. Let $c,\eta,\mu,i$ be parameters such that $0<c$, $1<\eta \leq 2$, and $0<i<\mu$. Define $r\in (i,\mu)$ and $l\in (-\mu,i)$ such that $$2(\mu-r)=c\eta (r-i)^{\eta-1}$$ and $$2(l+\mu)=c\eta (i-l)^{\eta-1}.$$ It is easy to see that $r$ and $l$ are uniquely defined within the given ranges. Then, I would like to prove that $$r^2+c(r-i)^{\eta} \leq l^2+c(i-l)^{\eta}.$$ In this problem, $i$ is the status-quo policy, $c$ and $\eta$ determine the cost of moving policies, $\mu$ is the degree of polarization of political parties, and $r$ ($l$) is the policy that a right-wing (left-wing) candidate would choose if elected. The inequality that I want to prove then simply says that the utility that the median voter will derive if $r$ is elected is at least as much as the one he/she will derive if $l$ is elected. We can assume that $r$ is the incumbent. If $\eta=2$, the result is straightforward. Together with my coauthors, I have already spent some time (in vain) trying to prove the claim. I have the impression that there has to be some easy argument, but in all proof strategies I have tried expressions turn extremely long and cumbersome immediately. By simulations, I also know that if $\eta>2$ or $i<0$ the result does not hold, so $1<\eta\leq2$ and $0<i<\mu$ should be used in the proof in some step. Any help or hint would be greatly appreciated. Thank you for your time!

It is my first time here. In a dynamic model of electoral competition ($t=1,2,\ldots$), I want to prove incumbency advantage, i.e., that the incumbent is always re-elected (unless an exogenous event occurs). It all boils down to the following result, which I am confident is true (all simulations work), but I have not been able to prove yet. Let $c,\eta,\mu,i$ be parameters such that $0<c$, $1<\eta \leq 2$, and $0<i<\mu$. Define $r\in (i,\mu)$ and $l\in (-\mu,i)$ such that $$2(\mu-r)=c\eta (r-i)^{\eta-1}$$ and $$2(l+\mu)=c\eta (i-l)^{\eta-1}.$$ It is easy to see that $r$ and $l$ are uniquely defined within the given ranges. Then, I would like to prove that $$r^2+c(r-i)^{\eta} \leq l^2+c(i-l)^{\eta}.$$ In this problem, $i$ is the status-quo policy, $c$ and $\eta$ determine the cost of moving policies, $\mu$ is the degree of polarization of political parties, and $r$ ($l$) is the policy that a right-wing (left-wing) candidate would choose if elected. The inequality I want to prove then simply says that the utility that the median voter derives if $r$ is elected is at least as much as the one he/she derives if $l$ is elected. We can assume that $r$ is the incumbent. If $\eta=2$, the result is straightforward. Together with my coauthors, I have already spent some time (in vain) trying to prove the claim. I have the impression that there has to be some easy argument, but in all proof strategies I have tried expressions turn extremely long and cumbersome immediately. By simulations, I also know that if $\eta>2$ or $i<0$ the result does not hold, so $1<\eta\leq2$ and $0<i<\mu$ should be used in the proof at some step. Any help or hint would be greatly appreciated. Thank you for your time!

It is my first time here. In a dynamic model of electoral competition ($t=1,2,\ldots$), I want to prove incumbency advantage, i.e., that the incumbent is always re-elected (unless an exogenous event occurs). It all boils down to the following result, which I am confident is true (all simulations work), but I have not been able to prove yet. Let $c,\eta,\mu,i$ be parameters such that $0<c$, $1<\eta \leq 2$, and $0<i<\mu$. Define $r\in (i,\mu)$ and $l\in (-\mu,i)$ such that $$2(\mu-r)=c\eta (r-i)^{\eta-1}$$ and $$2(l+\mu)=c\eta (i-l)^{\eta-1}.$$ It is easy to see that $r$ and $l$ are uniquely defined within the given ranges. Then, I would like to prove that $$r^2+c(r-i)^{\eta} \leq l^2+c(i-l)^{\eta}.$$ In this problem, $i$ is the status-quo policy, $c$ and $\eta$ determine the cost of moving policies, $\mu$ is the degree of polarization of political parties, and $r$ ($l$) is the policy that a right-wing (left-wing) candidate would choose if elected. The inequality that I want to prove then simply says that the utility that the median voter will derive if $r$ is elected is at least as much as the one he/she will derive if $l$ is elected. We can assume that $r$ is the incumbent. If $\eta=2$, the result is straightforward. Together with my coauthors, I have already spent some time (in vain) trying to prove the claim. I have the impression that there has to be some easy argument, but in all proof strategies I have tried expressions turn extremely long and cumbersome immediately. By simulations, I also know that if $\eta>2$ or $i<0$ the result does not hold, so $1<\eta\leq2$ and $0<i<\mu$ should be used in the proof in some step. Any help or hint would be greatly appreciated. Thank you for your time!