A Non-existence Theorem for Clientelism in Spatial Models

Abstract

This chapter proposes a spatial model that combines both programmatic as well as clientelistic modes of vote-seeking. In the model political parties strategically choose: (1) their programmatic policy position, (2) the effort they devote to clientelism as opposed to the promotion of their programmatic position, and (3) the set of voters who are targeted to receive clientelistic benefits. I present a theorem which demonstrates that, in its most general form, a spatial model with clientelism yields either Downsian convergence without clientelist targeting, or an inifinite cycle. Put otherwise, in its most general form the model never yields a Nash Equilibrium with positive levels of clientelism. I relate this result to past research on instability in coalition formation processes, and then identify additional restrictions, regarding voter turnout and the set of voters which parties can target, which serve to generate Nash equilibria with positive clientelist effort.

Theoretical Appendix

1.1 6.1 Proof of Lemma 2 for the Case

If and P’s opponent ∼P chooses v _m , it will be impossible to for P to persuade any voters on programmatic grounds. To see this note that, when , no voter will have a purely programmatic utility for P greater than (i.e. for all voters). As well, note that all voters have a programmatic utility of at least for any candidate ∼P who chooses v _m : the voters least satisfied with this platform are those with ideal points x _i =1 and x _i =0, and for these voters for any party ∼P which chooses the median voter programmatic vector v _m .

As a result, when and P’s opponent ∼P chooses v _m ,P will only gain the support of voters who are in its target set. In turn, any deviation from the outcome v ₁=v ₂=v _m will need to involve a target set of at least half the electorate in order to give P a chance of winning. Furthermore, any target set greater than a bare plurality contains more voters than necessary to win the election, and thus will not represent the necessary condition choices $\underline{\hat{x}}_{P}(G_{P})$, and $\hat{\overline{x}}_{P}(G_{P})$ (recall above definition of necessity).

By Assumption 1 above, this bare plurality target set will include the median voter. The median voter will be the voter from this target set whose allegiance will be most difficult to gain, since the opposing party ∼P chooses the median voter’s ideal point at v _m . It follows that $\hat{x}_{P}(G_{P}) = x_{m}$.

1.2 6.2 Lemma 3 and the Ideological Swing Voter

When and P’s opponent ∼P chooses v _m , it may be possible to for P to persuade some voters on programmatic grounds. In turn, there may exist payoff-enhancing deviations for P which do not involve choosing a bare plurality target set. Lemma 3 establishes the necessary condition strategy for a payoff-enhancing deviation which does not involve a bare plurality target set. Put otherwise, if the strategy identified in Lemma 3 leads does not lead to , then no deviation without a bare plurality target set is payoff-enhancing. Lemma 3 establishes the necessary condition strategy for a payoff-enhancing deviation on the political right; a symmetric condition applies on the political right.

Lemma 3

For any , the necessary condition strategy without a bare plurality target set on the political right is and .

This lemma, tells us that for any the necessary condition strategy for payoff-enhancing deviation on the political right involves the platform and the target set . For example, if G _P =.8 then $\hat{x}_{P}(.8) = .7$ and the C _P =.2 units of clientelistic effort will be targeted to voters in the range $\hat{\varTheta } _{P} = [.5,.7]$.

Proof of Lemma 3

When one party ∼P chooses the median-voter programmatic strategy vector v _m and her opponent P chooses x _P and , define x _S as the swing ideological voter, a voter whose programmatic utility for party P is the same as his or her programmatic for party ∼P:

$$ u_{S,P}(\mathrm{prog}) = u_{S,\sim P}(\mathrm{prog})\quad \Rightarrow\quad G_{P} \cdot\bigl(1 - abs[x_{P} - x_{S}]\bigr) = 1 - abs[x_{m} - x_{S}]. $$

(A.1)

We will now identify, for any , the swing ideological voter x _S when ∼P chooses v _m and P chooses , i.e. when P chooses an ideological deviation on the political right. An identical process applies for deviations on the political left. Note first that swing ideological voters may exist both in the range and in the range [x _P ,1], i.e. both voters to the left and to the right of x _P may be indifferent between the parties’ respective programmatic stances.¹⁰

Define $\underline{x}_{S}$ as a swing ideological voter in the range . Given our specification of programmatic utility u _i,P (prog), for any the following expression implicitly defines $\underline{x}_{S}$ when ∼P chooses v _m and P chooses :

(A.2)

This can be rewritten as:

(A.3)

Based on (A.3) I establish the following Sub-lemma:

Sub-lemma 1

For any , when ∼P chooses v _m and P chooses , there is no swing voter ideological voter $\underline{x}_{S}$ in the range for values of .

Proof of Sub-lemma 1

We are looking for swing ideological voters in the range . As such, if (A.3) generates a value $\underline{x}_{S} > x_{P}$, then there is no swing ideological voter $\underline{x}_{S}$ in the range . To see this, note that (A.2) above applies only to voters in the range . In turn, if (A.3) generates a value $\underline{x}_{S} > x_{P}$, we know that the indifference conditions for a swing voter in the range are not satisfied for voters in the applicable range, such that there is no swing voter ideological voter $\underline{x}_{S}$ in the range . It is then straightforward to establish that (algebra omitted), for any :

 □

In turn, for any Sub-lemma 1 allows to express $\underline{x}_{S}$ as follows:

(A.4)

We now move to identifying ideological swing voters $\overline{x}_{S}$ in the range [x _P ,1]. Given our specification of programmatic utility u _i,P (prog), for any the following expression implicitly defines $\overline{x}_{S}$ when ∼P chooses v _m and P chooses :

(A.5)

This can be rewritten as:

(A.6)

Based on (A.6) we can establish the following Sub-lemmas:

Sub-lemma 2

For any , when ∼P chooses v _m and P chooses , there is no swing voter ideological voter $\overline{x}_{S}$ in the range [x _P ,1] for values of .

Sub-lemma 3

For any , when ∼P chooses v _m and P chooses , there is no swing voter ideological voter $\overline{x}_{S}$ in the range [x _P ,1] for values of .

Proof of Sub-lemma 2

We are looking for swing ideological voters in the range [x _P ,1]. By definition, if (A.6) generates a value $\overline{x}_{S} > 1$, then there is no swing ideological voter $\overline{x}_{S}$ in the range [x _P ,1]: no voters in the applicable range satisfy the indifference condition in (A.6). It is then straightforward to establish that (algebra omitted):

 □

Proof of Sub-lemma 3

We are looking for swing ideological voters in the range [x _P ,1]. By definition, if (A.6) generates a value $\overline{x}_{S} < x_{P}$, then there is no swing ideological voter $\overline{x}_{S}$ in the range [x _P ,1]: no voters in the applicable range satisfy the indifference condition in (A.6). It is then straightforward to establish that (algebra omitted),

 □

Sub-lemmas 2 and 3 allow us to express $\overline{x}_{S}$ as follows:

(A.7)

Taken together, expressions (A.4) and (A.7) tell us that, for any , when ∼P chooses v _m and P chooses the game never has more than one swing voter, i.e. the existence conditions stipulated in Sub-lemmas 1, 2, and 3 are never simultaneously satisfied for both $\underline{x}_{S}$ and $\overline{x}_{S}$. Furthermore, they allow us to precisely identify the swing ideological voter for any and :

(A.8)

In words, when the game has no swing ideological voters. At such moderate values of x _P , all voters have a higher programmatic utility for party ∼P than for partyP, because the latter has not sufficiently distinguished her programmatic stance from the median voter policy adopted by ∼P. In contrast, at intermediate values of the game’s swing ideological voter will be $\overline{x}_{S} \in[x_{P},1]$, and the subset of extremist voters in the range $[\overline{x}_{S},1]$ will have a higher programmatic utility for P than for ∼P despite the fact that G _∼P =1>G _P . Finally, at more extreme values of , the game’s swing ideological voter will be , and all voters in the range $[\underline{x}_{S},1]$ will have a higher programmatic utility for P than for ∼P despite the fact that G _∼P =1>G _P .

Note from the above swing voter analysis that, for any value of , voters with ideal points in the range [x _S ,1] have a higher programmatic utility for party P than for party ∼P. It follows immediately from (A.8) that, for any , the programmatic position is the position which maximizes the range of [x _S ,1], i.e. maximizes the number of voters who prefer P on purely programmatic grounds. For any and , P will only target clientelistic goods to some subset of voters with ideal points x _i <x _S , since those with ideal points x _i >x _S can be counted on to choose P on purely programmatic grounds. It follows that the necessary condition strategy given some includes the platform : this is the policy position which maximizes the number of P’s ideological supporters, and in turn minimizes the size of Θ _P to which P’s clientelistic efforts will need to be targeted so as to secure a bare majority.

When P chooses , it is straightforward to see from (A.8) above that the game’s swing ideological voter has ideal point , i.e. that the swing ideological voter is the voter whose ideal point is identical to P’s programmatic position. All voters with ideal points prefer ∼P to P on purely programmatic grounds, and vice versa for voters with ideal points . In turn, given that we know that , i.e. that target set most conducive to securing a bare majority victory, is that which targets all voters between the median ideal point and the swing voter . □

1.3 6.3 Proof of Lemma 2 for the Case

The median voter receives a utility of ‘1’ from the set of actions v _m . On the other hand, Lemma 2 tells us that, when η=1, the median voter’s utility for necessary condition deviations when will be:

(A.9)

When , party P can consider both locally optimal deviations with a bare majority is target set and the median policy stance (Lemma 2), or deviations to the political right or left (Lemma 3). If the former, the median voter’s utility when η=1 will be (A.9). If the latter, the median voter’s utility for locally optimal deviations when η=1 will be:

$$ u_{m,P}\bigl(\hat{x}(G_{P},),\hat{\varTheta } _{P}(G_{P})\bigr) = (G_{P})^{2} + \biggl( \frac{1 - G_{P}}{\delta+ 1 - G_{P}} \biggr) . $$

(A.10)

To prove Lemma 2, I first establish that, for any , the median voter will always receive a higher utility from the deviation stipulated in Lemma 2 than that stipulated in Lemma 3: (A.9) > (A.10) (algebra omitted). This in turn implies that the strategy identified Lemma 2 is more likely to yield payoff-enhancing deviations than is that identified in Lemma 3, i.e. if the strategy from Lemma 2 yields a payoff-enhancing deviation then so does the strategy in Lemma 3, but not vice versa. This establishes Lemma 2 in the text, i.e. that for any value of G _P <1 Lemma 2 identifies the necessary condition strategy for payoff-enhancing deviations.

1.4 6.4 Proof of Proposition 1

When η=1, as long as there does not exist a payoff-improving deviation from v _m to a value G _P <1, and conversely as long there does exist a payoff-improving deviation from v _m to a value G _P <1.

Given a deviation from v _m to the necessary condition strategy, it is straightforward to see that, as long as the median voter prefers the deviating candidate P to the her opponent ∼P, then do all other voters in P’s target set. The median voter receives a utility of ‘1’ from the set of actions v _m . On the other hand, when η=1, the median voter’s utility for the necessary condition strategy when G _P <1 will be:

(A.11)

In turn it is straightforward to see that, for values of G _P <1, the function can only be greater than ‘1’ if (algebra omitted).