## _wp07165

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---

### Major themes and research question
- Vote buying is a common form of corruption and an important obstacle to welfare-enhancing policies and economic growth.
- Central research question: how do size, secrecy, and contractual sophistication affect the buyability of voting bodies, and how do these effects depend on whether there is competition among interest groups seeking to influence voting outcomes.
- Approach: reexamine the model of Groseclose and Snyder (GS, 1996) under variations in the contractual environment (who can contract on what contingencies) and under changes in size and ballot secrecy.

### Historical and illustrative evidence (preserved examples)
- 1757: George Washington defeated 270 to 40 on a temperance platform; 1758: he offered an average of one and a half quarts of alcoholic beverages and won by 310 to 45; total electorate ≈ 350 in those elections.
- 19th Amendment in 1920 extended suffrage to women; U.S. House expanded from 65 (First Congress of 1789) to 435; U.S. Senate expanded from 26 to 100.
- 1856: Victoria (Australian state) first to adopt the secret ballot.
- 2000 Taiwan: ruling National Party subsidized betting parlors with favorable odds contingent on electoral outcome to circumvent ballot secrecy.
- 2002 Salt Lake City Olympic scandal: IOC members reportedly paid money per vote and a "bonus" conditional on outcome.
- 1830s Liverpool: reported vote-price fluctuations likened to a stock price (Seymour, 1915, p.167).

### Model, structure, and scope
- Voting body: odd number, n, of voters choosing between policies a and b under majority rule.
- Interest groups: A (prefers a) and B (prefers b); group payoffs WA > 0 when a is adopted (zero otherwise); WB > 0 when b is adopted (zero otherwise).
- Voters i = 1,...,n with payoffs Ui(ci, ti) = ui(ci) + ti; define vi = ui(ci = a) − ui(ci = b); assume vi ≠ 0 and indexed so vi strictly decreases in i.
- Median voter M ≡ (n+1)/2 and assume vM < 0 (median prefers b absent vote buying).
- Contracting/enforcement: interest groups can offer enforceable, self-enforcing contracts contingent on observables (votes, outcomes, vote shares); extensive form: A offers first, B observes and may respond, voters choose one contract and vote.
- Definitions:
  - Successful vote buying contract: guarantees adoption of a in all subgame perfect equilibria following the contract.
  - Coalition for a: voters who weakly prefer to accept A’s contract and vote for a.
  - Winning coalition: coalition with cardinality at least M.
  - Outside option for a voter in B’s coalition: payoff the voter would receive if he unilaterally accepted A’s contract.

### Preliminaries — GS discriminatory vote buying recapitulation
- Fix coalition size m with M ≤ m ≤ n.
- K(m) = WB / (m − M + 1).
- K(m) contract (for v−1(K(m)) ≤ i ≤ m):
  - ti = K(m) − vi if ci = a;
  - ti = 0 if ci = b;
  - null contract for i < v−1(K(m)) or i > m.
- Cost: CA(m) = sum_{i=v−1(K(m))}^m ti.
- Proposition 1: Let m̃ ∈ arg min CA(m). Then a K(m̃) contract is a least-cost successful contract under discriminatory vote buying.
- Key GS insight: optimal to buy a supermajority (m̃ > M) because increasing m can lower per-voter surplus K(m) and deter B at lower total cost.

### A. The Size of the Voting Body
- Direct effect: if cost per vote is fixed, expanding n makes vote buying more costly (more voters to bribe).
- Strategic effect: competition among buyers enables buying a supermajority that reduces per-voter deterrence K(m); this can make larger bodies easier (cheaper) to buy.
- Net effects:
  - Absent competition: increasing size provides effective protection against vote buying.
  - With competition: larger voting bodies may be more buyable than smaller ones.
- Scaling framework: v(·) continuous on [0,1]; voter i’s preference vi = v((i−1)/(n−1)); median v(1/2) independent of n, assume v(1/2) < 0.
- Illustrative numerical example:
  - Set WB = 1 and
    - v(x) = 1/3 if x < 0.5
    - v(x) = −1/1000 if 0.5 ≤ x ≤ 0.6
    - v(x) = −2 if x > 0.6
  - Least-cost contract: A bribes all supporters of a plus up to three moderates; never bribes strong b supporters (vi = −2).
  - As n increases, costs jump when number of moderates in coalition increases by one; strategic effect can cause cost drops when moving to larger n at those jumps.
- Proposition 2: It can be cheaper for group A to bribe a larger voting body than it is to bribe a smaller voting body.

### B. The Secret Ballot (outcome-contingent contracting)
- Secret ballot modeled as making individual votes unobservable so only outcomes (or aggregate measures) are contractible.
- Empirical circumvention: forged ballots, absentee ballot purchase, negative vote buying, betting/subsidy schemes contingent on outcomes.
- Proposition 3:
  - If vM−1 < 0 (policy b enjoys supermajority intrinsic support), then successful vote buying contracts do not exist when only outcomes are contractible.
  - Intuition: with intrinsic supermajority for b, no voter is pivotal; outcome-based incentives cannot induce deviation.
- Proposition 4:
  - When only outcomes are contractible, a K(M) contract is a least-cost contract such that there exists an equilibrium in which policy a is adopted.
  - If vM−1 > 0 (policy a has intrinsic simple majority support), then a K(M) contract is a least-cost successful contract.
- Corollary 1: It is always cheaper for A to contract on votes than to contract on outcomes.
- Comparative implication: secret ballot (outcome-only contracting) is more effective at preventing vote buying when interest groups compete; if A is sole buyer, secret ballot may not raise the cost of a successful scheme (cost if successful can equal open-ballot cost), though success may be less likely.

### Complexity of contracts — three contract-complexity variations
1. Payments contingent on individual votes and policy outcomes.
2. Payments contingent on individual votes and vote shares.
3. Pure non-discriminatory (identical across voters) contingent payments.

A. Contracting on Votes and Outcomes
- Proposition 5:
  - When contracts can be contingent on both votes and outcomes, then a K(m̃) contract is a least-cost successful contract.
- Interpretation: conditioning additionally on outcomes does not reduce least-cost below the discriminatory vote-only benchmark because outcome contingencies are exploitable by the opponent B; A must pay as in vote-only K(m̃) contract to preclude B recruiting a supermajority.

Contracting on Votes and Vote Shares
- Proposition 6 (least-cost successful contract when contracts can be contingent on votes and vote shares):
  - For v_{-1(K(M+ 1))} ≤ i ≤ M+ 1:
    - t_i = max(-v_i; 0) if c_i = a and #a ≥ M+ 1
    - t_i = K(M+ 1) − v_i if c_i = a and #a < M+ 1
    - t_i = 0 if c_i = b
  - Null contract for i < v_{-1(K(M+ 1))} or i > M+ 1.
- Mechanism: A must deter B from recruiting either a bare majority or a supermajority; promises ensure each recruited voter gets surplus K(M+ 1) in deterrence events.
- Equilibrium path: A recruits a supermajority and compensates voters only for disutility of voting against their intrinsic preference; recruited voters receive zero net surplus when part of a winning supermajority.
- Corollary 2: The least-cost successful contract when B is present costs A the same amount as when B is absent but M+ 1 votes are required for passage of a.
- Policy implication: contracts contingent on votes and vote shares make vote buying extremely cheap even with competition; such contracts must be made extremely costly to deter (e.g., forfeiture of office or heavy fines).

Buying out the Competition
- If A buys out B first by offering W_B + " (arbitrarily small ") to B, total cost:
  - C_A = sum_{i=0}^M min(-v_i; 0) + W_B + "
- Cost under a least-cost successful contract that conditions on votes and vote shares:
  - C'_A = sum_{i=0}^M min(-v_i; 0) − v_{M+1}
- Remark 1: Under mild restriction that B cares more about b than voter M+ 1 cares about his vote for b, when contracts can be contingent on votes and vote shares, it is cheaper for A to contract only with voters than to buy out B.

Real-world illustration: Olympic bid example where payments of $500,000 to $1 million per IOC member plus outcome-contingent bonus of $3–5 million are reported.

### Non-Discriminatory Vote Buying
- Uniform transfer t_A; let m(t_A) be highest index i with v_i + t_A ≥ 0.
- Proposition 7 (least-cost successful contract under non-discriminatory vote buying):
  - t_A = W_B / M − v_M.
  - Group B offers the null contract.
  - All voters with i ≤ m(t_A) are in A’s coalition and vote for A.
- Transfers decompose:
  1. Compensatory payment offsetting intrinsic preferences:
     - Discriminatory: compensatory payments −v_i vary by voter.
     - Non-discriminatory: compensatory payment = −v_M.
  2. Surplus payment to deter B:
     - Discriminatory: surplus = W_B / (m' − M + 1).
     - Non-discriminatory: surplus = W_B / M.
- Proposition 8: Under non-discriminatory vote buying and Assumption 1, A always buys a supermajority in a least-cost successful contract.
- Rationing extension: allowing rationing (offering null to some voters) does not change the conclusion that buying a supermajority is optimal (see Proposition 9 in Appendix).

### Which is Less Costly: Discriminatory or Non-Discriminatory?
- Without competition: discriminatory vote buying is cheaper than non-discriminatory (compensatory payments are smaller; surplus payments zero).
- With competition: ambiguity—non-discriminatory may be cheaper because surplus payments W_B / M are smaller; discriminatory ability also allows B to target the weakest links in A’s coalition.
- Dominance depends on the shape of v_i around the median:
  - If many voters left of median intrinsically prefer a, discrimination cheaper.
  - If intrinsic support left of median is weak and deterrence costs under discrimination are large, non-discriminatory cheaper.

### Related literature
- Builds on Groseclose and Snyder (1996).
- Related works: Dal Bo (2004), Dekel, Jackson, and Wolinsky (2005), Meyerson (1993), Tullock (1972, 1980), Grossman and Helpman (1994, 1996, 1999), Bernheim and Whinston (1986), and others cited in the source.

### Conclusions and policy implications
- Key findings:
  - Increasing the size of a voting body raises the cost of successful vote buying when only a single interest group is present.
  - With competing interest groups, larger voting bodies may be cheaper to buy.
  - The secret ballot (modeled as outcome-contingent contracting) raises the cost of successful vote buying in the presence of competition; with a single interest group the secret ballot is much less effective.
  - Contractual complexity matters:
    - Adding outcome contingencies to vote-based contracts does not improve A’s position.
    - Contracts contingent on vote shares plus individual votes can make vote buying extremely cheap even with competition.
  - Non-discriminatory vote buying can be less costly than discriminatory vote buying when interest groups compete; the reverse holds absent competition.
- Policy implications:
  - Presence of competition among interest groups does not guarantee that outcomes reflect voter preferences or that vote buying costs increase.
  - Sophisticated contracts can substantially nullify competition.
  - Anti-influence tools (e.g., secret ballot, franchise extension) have effectiveness that depends crucially on the contracting environment and presence/absence of competition.
  - Example implications:
    - Extending the franchise can perversely make it cheaper for interest groups to influence outcomes, but only when interest groups compete.
    - The secret ballot is a robust deterrent when there is competition but less helpful when competition is absent.

### Appendix I — Proofs of key propositions and conditions (overview)
- Proofs establish:
  - Proposition 3: outcome-only contracting fails when b has intrinsic supermajority.
  - Proposition 4: K(M) is least-cost outcome-only contract when a can be induced.
  - Proposition 5: K(m') is least-cost even when contracts can condition on votes and outcomes.
  - Proposition 6: construction and cost characterization when contracts can condition on votes and vote shares; least-cost recruits minimal supermajority.
  - Proposition 7 & 8: least-cost non-discriminatory contract t_A = W_B / M − v_M and supermajority optimality under Assumption 1.
  - Proposition 9 (rationing extension): under Assumption 2 (v_i − v_{i+1} ≤ W_B / M for all i) supermajority remains least-cost with rationing.
- Cost comparisons and deterrence inequalities used throughout (exact expressions preserved in main text proofs).

*Excerpted from the provided IMF working paper content.*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   26

### _wp07165 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   26

### Major themes and research question
- Vote buying is a common form of corruption and an important obstacle to welfare-enhancing policies and economic growth.
- Central research question: how do size, secrecy, and contractual sophistication affect the buyability of voting bodies, and how do these effects depend on whether there is competition among interest groups seeking to influence voting outcomes?
- The paper reexamines the model of Groseclose and Snyder (GS, 1996) to study these issues under variations in the contractual environment.

### Historical and illustrative evidence (preserved examples)
- 1757: George Washington ran for the Virginia House of Burgesses on a temperance platform and was defeated by 270 to 40 votes.
- 1758: Washington changed tactics and offered voters an average of one and a half quarts of various alcoholic beverages; he won by 310 to 45 votes.
- Size context: total number of voters in Washington's elections was only about 350.
- 19th Amendment in 1920 extended suffrage to women, contributing to larger electorates.
- U.S. legislative expansion: House of Representatives now numbers 435 members, whereas it had only 65 at the time of the First Congress of 1789; U.S. Senate expanded from 26 members to its current total of 100.
- 1856: the Australian state of Victoria was the first to adopt the secret ballot in general elections.
- 2000: Presidential election in Taiwan Province of China — ruling National Party subsidized betting parlors with favorable odds contingent on electoral outcome to circumvent ballot secrecy.
- 2002: Salt Lake City Olympic Winter Games scandal — IOC members reportedly paid money for votes and a "bonus" conditional on the outcome (success of the city’s Olympic bid).
- 1830s Liverpool elections: reported vote-price fluctuations described like a stock price (Seymour, 1915, p.167).

### Core analytical findings (model-based)
- Direct effect of increasing the size of the voting body: if cost per vote is fixed, expanding the voting body makes vote buying more costly.
- Strategic effect under competition: competition among vote buyers can reduce per-voter deterrence costs because optimal strategies (per GS, 1996) include buying a supermajority to blunt competition; the magnitude of the optimal supermajority varies with size and may allow economizing on deterrent payments as size grows.
- Net effect of size:
  - Absent competition: increasing size provides effective protection against vote buying.
  - In presence of competition: larger voting bodies may be more buyable than smaller ones.
- Effect of secret ballot:
  - Absent competition: secret ballot has little effect on the cost of vote buying; it does not affect cost but may reduce likelihood via equilibrium multiplicity.
  - Presence of competition: secret ballot unambiguously increases the cost of vote buying and decreases the likelihood relative to the GS case where votes are directly contractible.
- Effect of discriminatory vs. non-discriminatory bribery:
  - Discriminatory vote buying (payments tailored to individual preferences) is always cheaper than non-discriminatory vote buying when there is no competition.
  - In the presence of competition, discriminatory buying need not be cheaper; conditions are identified where the legislature is more buyable under non-discriminatory contracts than under discriminatory contracts.
- Effect of contracting on votes versus outcomes:
  - Ability to contract on votes and outcomes has no effect on the cost of vote buying compared to contracting only on votes; this holds independent of competition.
  - However, contracting on votes and vote shares (aggregate measures) changes this: when interest groups can contract on votes and vote shares, vote buying becomes extremely cheap even with competition, leaving the voting body uniquely at risk of "capture."

### Model, structure, and scope
- The model used is exactly that of GS (1996) except for variations in the contractual environment (size, secrecy, and sophistication).
- Paper roadmap:
  - Section 2: model description.
  - Section 3: recapitulation of main result of GS model (characterization of the optimal discriminatory vote buying contract).
  - Section 4: effects of size and secrecy on buyability.
  - Section 5: effects of contractual complexity/sophistication on buyability.
  - Section 6: results in context of broader literature.
  - Section 7: conclusion.

### Figures referenced in the content
- Figure 1: Cost of Discriminatory Vote Buying as a Function of n.
- Figure 2: Cheap Discriminatory Vote Buying.
- Figure 3: Cheap Non-Discriminatory Vote Buying.

*Source: Content unit _wp07165 - References . . . . . . . . . . . . . . . . . . . . . . . . .   26*

### Appendix A contains proofs of results presented in the main text, while Appendix B

### _wp07165 - Appendix A contains proofs of results presented in the main text, while Appendix B

### The Model
- Setup:
  - Odd number, n, of voters choosing between two policies labeled a and b; majority rule.
  - Two interest groups A and B: A prefers a, B prefers b.
  - Group payoffs: WA > 0 when a adopted (zero otherwise); WB > 0 when b adopted (zero otherwise).
  - Voters i = 1,2, ..., n have payoffs Ui(ci, ti) = ui(ci) + ti, where ci ∈ {a, b} and ti are transfers from interest groups.
  - Define vi = ui(ci = a) − ui(ci = b); assume vi ≠ 0 for all i and indices ordered so vi strictly decreases in i.
  - Median voter M ≡ (n+1)/2 and assume vM < 0 (median prefers b absent vote buying).
  - Define v−1(x) ≡ min{ i | vi ≤ x }.
- Contracting and enforcement:
  - Interest groups can offer enforceable contracts (bribes) to induce votes; enforcement assumed via reputational effects (self-enforcing).
  - Contracts may be contingent on different observables (votes, outcomes, vote shares); the paper varies which contingencies are allowed.
- Equilibrium and definitions:
  - Extensive form: A offers contracts to all voters; B observes and then offers contracts; voters choose one contract and vote; outcome determined.
  - Tie-breaking: If B cannot do better than null contract it chooses null; indifferent voters accept A’s contract.
  - Definition 1: A vote buying contract is successful iff it guarantees adoption of policy a (all subgame perfect equilibria following the contract lead to a).
  - Definition 2: A coalition for policy a consists of voters who weakly prefer to accept A’s contract and vote for a, given offers.
  - Winning coalition: coalition with cardinality at least #M.
  - Definition 3: Outside option for a voter in B’s coalition is the payoff the voter would receive if he unilaterally accepted A’s contract.

### Preliminaries (discriminatory vote buying — Groseclose and Snyder (GS) recapitulation)
- Fix coalition size m with M ≤ m ≤ n.
- Define K(m) as the minimum expected payoff earned by any voter i = 1,..., m (payoffs include transfers) under A’s scheme chosen so B will not re-bribe. For B to obtain b it must re-bribe m − M + 1 voters, so:
  - K(m) = WB / (m − M + 1)
- A K(m) contract (for v−1(K(m)) ≤ i ≤ m) sets transfers:
  - ti = K(m) − vi if ci = a
  - ti = 0 if ci = b
  - null contract for i < v−1(K(m)) or i > m
- Cost of such a contract:
  - CA(m) = sum_{i=v−1(K(m))}^m ti
- Proposition 1:
  - Let m̃ ∈ arg min CA(m). Then a K(m̃) contract is a least-cost successful contract under discriminatory vote buying.
- Key GS insight summarized:
  - It is typically optimal for A to buy a supermajority (m̃ > M) because doing so reduces the per-voter surplus K(m) and can reduce total cost by deterring B.

### Policy Responses to Vote Buying
A. The Size of the Voting Body
- Scaling preferences with n:
  - Introduce v(·) continuous strictly decreasing on [0,1]; impose grid of size n (odd) so voter i’s relative preference vi = v((i−1)/(n−1)).
  - Median voter index i = (n+1)/2 has relative preference v(1/2) independent of n; assume v(1/2) < 0.
- Main qualitative result:
  - The strategic effect (benefit from buying a supermajority and lowering K(m)) can dominate the direct effect (more voters to bribe), so larger voting bodies can be more buyable than smaller ones.
- Illustrative example (numerical):
  - Set WB = 1 and preference function:
    - v(x) = 1/3 if x < 0.5
    - v(x) = −1/1000 if 0.5 ≤ x ≤ 0.6
    - v(x) = −2 if x > 0.6
  - Voters partition: supporters of a (x < 0.5), moderates (0.5 ≤ x ≤ 0.6), strong supporters of b (x > 0.6).
  - Result: least-cost contract has A bribing all supporters of a plus up to three moderates; never bribes strong b supporters (vi = −2).
  - As n increases, costs jump at points when number of moderates in coalition increases by one; strategic effect can cause cost to drop when moving to larger n at those jumps.
- Proposition 2:
  - It can be cheaper for group A to bribe a larger voting body than it is to bribe a smaller voting body.

B. The Secret Ballot
- Rationale and historical notes:
  - Secret ballot makes individual votes unobservable and aims to prevent contracting on individual votes.
- Circumvention strategies (empirical examples summarized):
  - Tasmanian Dodge, forged/filled ballots, absentee/permanent absentee ballots vulnerable to purchase, negative vote buying (paying opposition supporters not to vote).
- Outcome-contingent contracting (only outcomes are contractible):
  - Proposition 3:
    - If vM−1 < 0 (policy b enjoys supermajority intrinsic support), then successful vote buying contracts do not exist when only outcomes are contractible.
    - Intuition: whenever voters have supermajority intrinsic support for b, there exists equilibrium where voters ignore A’s contract and vote by intrinsic preferences — no one sees themselves as pivotal, so outcome-based incentives fail.
  - Proposition 4:
    - When only outcomes are contractible, a K(M) contract is a least-cost contract such that there exists an equilibrium in which policy a is adopted.
    - Furthermore, if vM−1 > 0 (policy b enjoys simple majority intrinsic support), then a K(M) contract is a least-cost successful contract.
  - Corollary 1:
    - It is always cheaper for A to contract on votes than to contract on outcomes.
- Comparative implications:
  - Outcome-only contracts blunt A’s ability to buy supermajorities; they force A toward buying a bare majority, which can be more expensive and reduces incentive for legislative capture.
  - The secret ballot is more effective at preventing vote buying when there is competition between interest groups than when A is the only group attempting to buy votes.
  - Note: introducing the secret ballot does not increase A’s cost if A can still succeed in the absence of competition — cost if successful can equal the open-ballot cost, but success may be less likely.

### Complexity
- Motivation:
  - Real-world contracts can be more complex (contingent on votes and outcomes; vote shares) or simpler (non-discriminatory payments).
  - Examples: Olympic bid scandal — payments $500,000 to $1 million per IOC member plus outcome-contingent bonus of $3–5 million.
- Three contract-complexity variations considered:
  1. Payments contingent on individual votes and policy outcomes.
  2. Payments contingent on individual votes and vote shares.
  3. Pure vote buying with non-discriminatory (identical across voters) contingent payments.
- A. Contracting on Votes and Outcomes
  - Proposition 5:
    - When contracts can be contingent on both votes and outcomes, then a K(m̃) contract is a least-cost successful contract.
  - Interpretation:
    - Allowing joint contingencies (votes and outcomes) does not reduce A’s least-cost of success below the discriminatory vote-only benchmark (K(m̃) contract).
    - Reason: although A could condition payments on outcomes to avoid paying in losses, such contracts are exploitable by B (B can recruit a supermajority at arbitrarily small cost if A withholds payments), so paying as in the vote-only K(m̃) contract is required to preclude B from recruiting a supermajority.
- Note on real-world deviations:
  - Outcome-contingent bonus schemes (e.g., Olympic example) may reflect budget constraints or other considerations; they do not necessarily correspond to least-cost successful contracts derived in the benchmark model.

*Source: _wp07165 - Appendix A contains proofs of results presented in the main text, while Appendix B (PDF chapter/section).*

### introduction of budget constraints creates substantial complications in the analysis (see, for

### _wp07165 - introduction of budget constraints creates substantial complications in the analysis (see, for

### Contracting on Votes and Vote Shares
- Proposition 6 (least-cost successful contract when contracts can be contingent on votes and vote shares):
  - For v_{-1(K(M+ 1))} ≤ i ≤ M+ 1:
    - t_i = max(-v_i; 0) if c_i = a and #a ≥ M+ 1
    - t_i = K(M+ 1) - v_i if c_i = a and #a < M+ 1
    - t_i = 0 if c_i = b
  - For i < v_{-1(K(M+ 1))} or i > M+ 1: the null contract is offered.
- Mechanism and intuition:
  - Group A must deter B from recruiting either a bare majority or a supermajority.
  - To deter a bare majority, A promises recruited voters generous payments when their votes are pivotal.
  - To deter a supermajority, A promises generous payments when recruited voters are part of a losing effort for A.
  - The contract promises each voter a surplus of K(M+ 1) under either deterrence event; this amount suffices to deter B.
  - On the equilibrium path, A recruits a supermajority and compensates voters only for disutility of voting against their preferred option; voters in A’s coalition receive zero net surplus when part of a winning supermajority.
  - The least-cost successful contract recruits the minimal supermajority because direct compensation costs (voters’ disutility from voting against their intrinsic preference) are minimized by the smallest supermajority; deterrence costs are off the equilibrium path and thus irrelevant for equilibrium cost minimization.
- Corollary 2:
  - The least-cost successful contract when B is present costs A the same amount as when B is absent but M+ 1 votes are required for passage of policy a.
- Policy implication from this section:
  - Contingent contracts that condition on votes and vote shares must be made extremely costly (e.g., penalties such as forfeiture of office or heavy fines) because they allow group A to largely nullify the effects of competition.

### Buying out the Competition
- Comparison of costs:
  - If A buys out B first, A can make a take-it-or-leave-it offer of W_B + " (arbitrarily small ") to B. Total cost to A:
    - C_A = sum_{i=0}^M min(-v_i; 0) + W_B + "
  - Under a least-cost successful contract that conditions on votes and vote shares, cost to A:
    - C'_A = sum_{i=0}^M min(-v_i; 0) - v_{M+1}
- Remark 1:
  - Under the mild restriction that interest group B cares more about policy b than individual voter M+ 1 cares about his vote for b, when contracts can be contingent on votes and vote shares, it is cheaper for group A to contract only with voters than to first buy out competing interest group B.
- Real-world illustration:
  - Historical anecdote of paying $7,000 to a candidate to quit a race for a less than $6,000-a-year Deputy’s job, illustrating feasibility and attractiveness of buying out competition in some settings.

### Non-Discriminatory Vote Buying
- Setup:
  - Let t_A be the uniform transfer offered by group A.
  - Let m(t_A) be the highest index i such that v_i + t_A ≥ 0.
  - If B offers the null contract, voters i = 1, ..., m(t_A) accept A’s t_A and vote accordingly.
- Proposition 7 (least-cost successful contract under non-discriminatory vote buying):
  - t_A = W_B / M - v_M.
  - Group B offers the null contract.
  - All voters with indices i ≤ m(t_A) are in A’s coalition and vote for A.
- Structure of transfers:
  - Transfers decompose into:
    1. Compensatory payment to offset intrinsic preferences favoring b:
       - Discriminatory case: compensatory payments are -v_i and vary by voter.
       - Non-discriminatory case: compensatory payment is -v_M (determined by the median voter).
    2. Surplus payment to deter B:
       - Discriminatory case: surplus payment equals W_B / (m' - M+1) (reflecting B’s ability to target selected voters).
       - Non-discriminatory case: surplus payment equals W_B / M (lower because B cannot target selected voters).
- Proposition 8:
  - Under non-discriminatory vote buying and Assumption 1, A always buys a supermajority of voters in a least-cost successful contract.
  - Intuition: If A bought only a simple majority, voter M would accept while voter M+ 1 would not; by Assumption 1 their intrinsic preferences are similar, so M’s net surplus under A’s contract would be close to zero, allowing B to invade with a tiny bribe. Therefore A must buy a supermajority.
- Note on rationing:
  - Adding rationing does not change the basic conclusion that buying a supermajority is optimal (see Appendix B referenced in source).

### Which is Less Costly: Discriminatory or Non-Discriminatory Vote Buying?
- Without competition:
  - Discriminatory vote buying is cheaper than non-discriminatory vote buying because compensatory payments are smaller under discriminatory schemes while surplus payments are zero.
- With competition:
  - Ambiguity arises because surplus payments are smaller under non-discriminatory vote buying, giving a potential advantage to non-discrimination.
  - Discriminatory ability has both advantages and disadvantages for A:
    - Advantage: A can avoid paying strongest supporters and pay weaker supporters less.
    - Disadvantage: B can discriminate and target the weakest links in A’s coalition.
  - Which effect dominates depends on the shape of the preference function v_i around the median voter.
- Graphical insights (as described in the source):
  - Figure 1 scenario: discriminatory vote buying cheaper when many voters to the left of the median intrinsically prefer a; discriminatory payments avoid paying many of these voters.
  - Figure 2 scenario: non-discriminatory vote buying cheaper when deterrence costs under discrimination are considerably higher and intrinsic support to the left of the median is weak, making discriminatory payments larger.

### Related Literature
- Builds on Groseclose and Snyder; extends their model in electoral system structure and contracting environment.
- Other relevant works:
  - Dal Bo (2004): contracts involving votes and vote shares without competition.
  - Dekel, Jackson, and Wolinsky (2005): contracts based on votes and outcomes; differs in that voters are non-strategic, interest groups are budget constrained, and vote buying is modeled as an alternating offer scheme.
- Connection to literature on electoral systems and corruption:
  - Meyerson (1993) examines how electoral systems differ in sorting corrupt vs. non-corrupt candidates.
  - Distinct from rent-seeking and influence-buying literature where voting plays little role; examples include Tullock (1972, 1980), Grossman and Helpman (1994, 1996, 1999), Bernheim and Whinston (1986).

### Conclusions and Policy Implications
- Key findings:
  - Increasing the size of a voting body increases the cost of successful vote buying when only a single interest group is present.
  - With competing interest groups, larger voting bodies may be cheaper to buy.
  - The secret ballot (modeled as forcing outcome-contingent rather than vote-contingent contracts) raises the cost of successful vote buying in the presence of competition; with a single interest group, the secret ballot is much less effective.
  - Contractual complexity matters:
    - Outcome-based contingencies added to vote-based contracts are worthless.
    - Contracts contingent on vote shares plus individual votes are extremely valuable and can make voting bodies uniquely vulnerable to being bought even with competition present.
  - Non-discriminatory vote buying can be less costly than discriminatory vote buying when interest groups compete; the reverse holds absent competition.
- Policy implications:
  - Presence of competition does not guarantee that policy outcomes reflect underlying voter preferences and does not necessarily raise the costs faced by interest groups.
  - Sophisticated interest groups can construct contracts that nearly nullify competition.
  - The effectiveness of anti-influence tools depends crucially on the contracting environment and whether interest groups compete.
  - Examples:
    - Extending the voting franchise can sometimes perversely make it cheaper for interest groups to influence outcomes, but this occurs only in the presence of competition.
    - The secret ballot is a robust deterrent in the presence of competition but is less helpful when competition is absent.
  - Policy makers must consider the contracting environment and the presence or absence of competition when designing anti-corruption measures.

*Italic source attribution: Excerpted from the provided IMF working paper content.*

### References

### References

### Bibliographic citations
- August, O. March 15, 2000. Betting alters the odds in close Taiwan election. The Times.
- Banks, J. 2000. Buying Supermajorities in Finite Legislatures. The American Political Science Review. 94: 677-681.
- Baron, D. (2000). "Legislative Organization with Informational Committees." American Journal of Political Science, 44 (3), pp. 485ñ505.
- Bernheim, B. D. and M. Whinston. 1986. Menu auctions, resource allocation, and economic ináuence. Quarterly Journal of Economics. 101: 1-31.
- Caro, R. 1982. The Years of Lyndon Johnson: The Path to Power. New York: Alfred A. Knopf.
- Gary W. Cox; J. Morgan Kousser. 1981. Turnout and Rural Corruption: New York as a Test Case. American Journal of Political Science. 25: 646-663.
- Dal Bo, E. 2004. Bribing voters. UC Berkeley Working Paper.
- Dekel, E., M. Jackson, and A. Wolinsky. 2005. Vote buying. Caltech Working Paper.
- Ford, Paul Leicester. 1896. The True George Washington. Philadelphia: J.B. Lippincott Co.
- Gilligan, Thomas W., and Keith Krehbiel. 1987. Collective Decision-Making and Standing Committees: An Informational Rationale for Restrictive Amendment Procedures. Journal of Law, Economics, and Organization 3(2):287-335.
- Gilligan, Thomas W., and Keith Krehbiel. 1989. "Asymmetric Information and Legislative Rules with a Heterogeneous Committee." American Journal of Political Science 33(2):459-490.
- Groseclose, T. and J. Snyder. 1996. Buying supermajorities. American Political Science Review. 90: 303-315.
- Grossman, G. and E. Helpman. 1994. Protection for sale. American Economic Review. 84: 833-850.
- Grossman, G. and E. Helpman. 1999. Competing for endorsements. American Economic Review. 89: 501-524.
- Grossman, G. and E. Helpman. 1996. Electoral competition and special interest politics. Review of Economic Studies. 63: 265-286.
- Heckelman, J. 1998. Bribing Voters Without VeriÖcation. The Social Science Journal. 35: 435-443.
- Morgan, John, and Felix V·rdy. 2007. Negative Vote Buying. mimeo.
- Nitzan, S. 1994. Modeling rent seeking contests. European Journal of Political Economy. 10: 41-60.
- Newman, T. 2003. Tasmania and the Secret Ballot, Australian Journal of Politics and History. 49: 93-101.
- Persson, T., G. Tabellini and F. Trebbi, 2003. Electoral Rules and Corruption. Journal of the European Economic Association. 1: 958-989.
- Quimpo, N. 2002. A season of vote buying and kidnappings. http://www.ipd.ph/features/july_2003/barangay_sk_elections.html
- Seymour, C. 1915. Electoral reform in England and Wales: The development and operation of the parliamentary franchise, 1832-1885. New Haven: Yale University Press.
- Sha§er, F. 2002. What is vote buying? Empirical evidence, in Vote Buying: Who, What, When and How? F. Sha§er and A. Schedler, eds. (forthcoming).
- Tullock, G. 1972. The purchase of politicians. Western Economic Review. 10: 354-55.
- Tullock, G. 1980. E¢ cient rent-seeking. In J.M. Buchanan, et al. (Eds.), Toward a theory of the rent-seeking society, College Station: Texas A&M Press.
- Webster, Hutton. 1920. Historical source book. Boston: D.C. Heath and Co.

### Appendix I — Proofs of Propositions: key results and conditions
- Proof of Proposition 3
  - For any contract offered by A, there exists a subgame perfect equilibrium where policy b is adopted.
  - Argument hinges on: A's contract contingent on outcomes is payoff relevant to voter i only if i can alter the outcome; if b commands a supermajority of intrinsic support then each voter has zero probability of affecting the policy by changing his vote; changing vote from b to a leads to a first order payoff effect of v_i; hence voters vote according to intrinsic preferences and B does nothing.

- Proof of Proposition 4
  - A cannot successfully buy a supermajority because then no member of that coalition would be pivotal; thus A must be buying a simple majority.
  - To deter B with a simple majority, the cost to B of recruiting a single member of A's coalition must be at least W_B; all members of A's coalition must obtain surplus of at least W_B when a is adopted.
  - All voters in A's coalition promised a positive payment when a is adopted must earn the same surplus; for such voters the transfer equals W_B - v_i.
  - An (outcome based) K(M) contract is a least-cost contract satisfying these properties.
  - Equilibrium characterization depends on v_{M-1}: if v_{M-1} < 0 then b is adopted when B does nothing; if v_{M-1} > 0 then voters perceive probability of being pivotal as one and accept A's K(M) contract.

- Proof of Proposition 5
  - For success, a contract must deter B from recruiting a majority of any size; to deter B from recruiting a bare majority, the outside options (joint event of voting for a and a winning) must sum to at least W_B.
  - The cheapest way to do this is a K(m') contract. Contradiction argument shows any cheaper contract fails to deter B.
  - K(m') contract also deters B from recruiting a supermajority by ensuring outside options (joint event of voting for a and losing) sum to at least W_B.
  - Conclusion: K(m') is a least-cost successful contract.

- Proof of Proposition 6
  - In equilibrium, cost of the contract is sum_{i=1}^{M+1} max(-v_i;0).
  - In absence of competition, minimum cost of obtaining #a = m votes is C_A(m) = sum_{i=1}^m max(-v_i;0).
  - Successful contracts that attempt #a = M are more costly because to deter B from re-bribing one voter, all voters for A must receive surplus at least W_B in event A is approved with exactly M votes.
  - The contract described has lower cost than those alternatives, and deters profitable deviations by voters and by B because any attempt by B to recruit k ≥ 2 voters from A's coalition requires payments k K(M+1) ≥ W_B for k ≥ 2.

- Proof of Proposition 7
  - Given A's offer, B's unique best response is the null contract because obtaining b would require recruiting up to the median voter M and offering at least t_B = W_B / M + " ; any successful contract for B costs at least C_B = W_B + M " which strictly exceeds B's payoff, so B prefers the null contract.
  - A's offer is successful and least-cost: if A offered t_A' < t_A then by offering t_B = t_A' - v_M + " group B could attract all voters with indices i ≥ M and for sufficiently small " this contract would cost B less than W_B, contradicting success.

- Proof of Proposition 8
  - For a supermajority of voters to receive payments from A, voter M+1 must be in A's coalition, i.e. v_{M+1} + t_A ≥ 0.
  - Substituting for t_A yields v_{M+1} - v_M + W_B / M ≥ 0.
  - Noting v_{M+1} < v_M < 0, rewrite as v_M - v_{M+1} ≤ W_B / M, which holds by Assumption 1.

### Rationing extension and Proposition 9
- Model amendment: groups restricted to offering either the null contract or a contract where t is a fixed transfer identical across constituents; rationing implemented by offering null contracts to some voters.
- Assumption 2: For all i, v_i - v_{i+1} ≤ W_B / M.
- Proposition 9 (non-discriminatory vote buying, rationing allowed)
  - Under a least-cost successful contract with rationing, A always buys a supermajority of voters.
  - Cost comparison:
    - If A recruits a simple majority (i = 1,...,M), each must enjoy payoff ≥ W_B; cost C_A^M = (W_B - v_M) (M - v_{-1} (W_B) + 1) [as stated in source text].
    - If A recruits a supermajority (i = 1,...,M+1), each must enjoy payoff ≥ W_B / 2; cost C_A^{M+1} = [ (W_B / 2 - v_M) ((M+1) - v_{-1} (W_B/2) + 1 ) ] [as stated in source text].
  - From Assumption 2 it follows that C_A^{M+1} < C_A^M, hence supermajority is least-cost.

*Italic: Source PDF filename: _wp07165 - References*

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