## _wp0611

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---

### I. Introduction — context and examples of vote buying
- Historical and contemporary instances:
  - George Washington (1757) offered constituents "an average of one and a half quarts of various alcoholic beverages" in exchange for votes.
  - In the 1996 Thai general elections "fully 30 percent of heads of households were offered money in exchange for their votes."
  - 19th-century Maryland: negative vote buying (payment for not voting).
  - Post-1872 England: material incentives contingent on aggregate turnout after secret ballot.
  - 2000 Taiwan Presidential election: ruling party subsidized betting parlors to offer favorable odds conditional on victory.
  - IOC scandal: members reportedly paid an up-front amount plus a "bonus" conditional on the vote outcome.
  - 1830s English boroughs: price of a vote varied over election day, increasing as the day progressed and falling once victory margin judged sufficient.
- Contract variants documented:
  - Payments for an individual vote.
  - Negative vote buying (payments for not voting).
  - Payments contingent on aggregate outcomes (turnout or election outcome).
  - Payments contingent on vote share or timing.
  - Indirect schemes (subsidizing third parties).

### II. Model — environment, timing, and objectives
- Environment and preferences:
  - Odd number n of voters choose between policies a and b; majority wins.
  - Two interest groups A and B prefer a and b respectively; payoffs W_A > 0 for A if a adopted, W_B > 0 for B if b adopted.
  - Voter utility: U_i(c_i, p, t_i) = u_i(c_i, p) + t_i.
  - Two polar specifications for u_i:
    - Vote-based motives: u_i independent of p; v_i = u_i(c_i = a) − u_i(c_i = b).
    - Outcome-based motives: u_i independent of c_i; v_i = u_i(p = a) − u_i(p = b).
  - v_i common knowledge, v_i ≠ 0 for all i, v_i strictly decreasing in i. Median voter M = (n+1)/2 with v_M < 0 (b would win absent A).
- Timing (extensive form):
  - A proposes a contract; after observing A's proposal, B may propose a contract; then votes are cast and payoffs realized.
  - Definition 1: A vote buying contract is successful iff it guarantees adoption of policy a (all subgame perfect equilibria following the contract lead to adoption of a).

### III. Overview of contractual variations analyzed
- Three contractual environments compared:
  - Contracts contingent only on votes.
  - Contracts contingent only on outcomes.
  - Contracts contingent on votes and vote shares.
- Benchmark: identical to Groseclose and Snyder (GS): individual votes contractible and voters have vote-based motives.
- Multiplicity: equilibria multiplicity can arise when contracts create interactive incentives or when voters have outcome-based motives; success uses conservative Definition 1.

### III.A Contracting on votes (details and key propositions)
- Vote-based motives (benchmark, GS case):
  - Monopoly (A alone) offers t_i = −v_i to all i ≤ M with v_i < 0.
  - Monopoly vote buying cost:
    - C^0_A = Σ_{i=1}^M max{0, −v_i}
  - With competition, fix coalition size m with n ≥ m ≥ M. Choose K so B will not invade:
    - C_B(K, m) = (m − M + 1) K
    - Require C_B(K, m) ≥ W_B → K(m) = W_B / (m − M + 1)
  - Define v^−1(K(m)) = lowest index i such that v_i < K(m). A's K(m)-contract (for coalition size m):
    - For v^−1(K(m)) ≤ i ≤ m:
      - t_i = K(m) − v_i if c_i = a
      - t_i = 0 if c_i = b
    - For i < v^−1(K(m)) or i > m: no contract offered.
  - Cost:
    - C_A(m) = Σ_{i=v^−1(K(m))}^m t_i
  - A chooses m to minimize C_A(m); denote minimizer m^*. A participates only if C_A(m^*) < W_A.
  - Proposition 1: A K(m^*) contract is a least-cost successful contract under vote-based motives where only individual votes are contractible.
  - Cost characterization:
    - C_A(m^*) = C^M_A + W_B + { (M − v^−1(K(m^*))) / (m^* − M + 1) W_B − Σ_{i=v^−1(K(m^*))}^{v^−1(0)−1} v_i } + Σ_{i=M+1}^{m^*} (−v_i)
    - The expression in curly brackets is positive; hence C_A(m^*) > C^M_A + W_B.

- Outcome-based motives (only votes contractible):
  - Contracts are more costly than in vote-based benchmark.
  - To deter B recruiting a supermajority, A must offer m − M voters transfers of at least
    - ϕ(m) = W_B / (m − M + 2) = K(m+1)
  - Candidate least-cost successful contract recruiting m voters:
    - For i ≤ m:
      - t_i = max{K(m+1), K(m) − v_i} if c_i = a
      - t_i = 0 if c_i = b
    - For i > m: no contract offered.
  - Define z(m) largest integer with K(m+1) ≥ K(m) − v_{z(m)}.
  - Cost:
    - C_A(m) = z(m) K(m+1) + Σ_{i=z(m)+1}^m (K(m) − v_i)
  - Let m^^ minimize C_A(m); choose largest integer if multiple.
  - Proposition 2: The contract with m = m^^ described above is a least-cost successful contract under outcome-based motives where only individual votes are contractible. This contract is more costly than the least-cost successful contract under vote-based motives where only individual votes are contractible.
  - Coalition size comparisons:
    - No unambiguous ranking of m^* and m^^ in general.
    - Remark 1: If v_i = β − γ(i − M) and n sufficiently large, then m^^ > m^* (larger fraction bribed under outcome-based motives).

- Empirical alignment:
  - Least-cost successful contract resembles documented vote buying strategies where bribes differ among supporters, undecided, and opposition, sometimes with undecided receiving up to three times supporters' amounts.

### III.B Contracting on outcomes (only outcome-contingent transfers)
- Motivation: historical secret-ballot responses included outcome/turnout contingent payments.
- Setting: transfers can be conditional on policy outcome only; transfers conditional on unfavored policy never optimal.
- Vote-based motives when only outcomes contractible:
  - Intuition that outcome contracts simplify bribery is incorrect.
  - Impossibility:
    - Proposition 3: If v_{M−1} < 0, then successful vote buying contracts do not exist under vote-based motives where only outcomes are contractible.
    - Reason: If b commands a supermajority absent vote buying, each voter has infinitesimal pivot probability; voting follows intrinsic preferences and A cannot change outcome via outcome-contingent payments.
  - If intrinsic support for b is only a bare majority, successful vote buying may be possible but buying a supermajority is impossible when only outcomes are contractible.
  - Proposition 4: If v_{M−1} > 0, then the K(M) contract is a least-cost successful contract under vote-based motives where only outcomes are contractible. K(M) induces a bare majority and is least-cost in this domain.

- Outcome-based motives when only outcomes contractible:
  - Proposition 5: The K(m  ) contract is a least-cost successful contract under outcome-based motives where only outcomes are contractible. Furthermore, the costs equal those when votes are contractible and voters have vote-based motives.

### III.C Contracts on votes and vote shares (enriched contract space)
- Notation: #a = number of votes cast for a; #b = n − #a.
- Vote-based motives with vote-share contingencies:
  - Competition has virtually no effect; least-cost successful contract costs A an amount C0A - v(M+ 1).
  - Proposition 6 (constructive contract): For v^−1(K(M+ 1)) ≤ i ≤ M+ 1, transfers specified as:
    - ti =
      - max ( -vi ; 0 ) if ci = a and #a ≥ M+ 1
      - K(M+ 1) − vi if ci = a and #a < M+ 1
      - 0 if ci = b
    - For i < v^−1(K(M+ 1)) or i > M+ 1: no contract offered.
  - Intuition: compensate loyal voters when a is at risk; minimal payments when a has supermajority; deters B from entering because recruiting k ≥ 2 voters requires paying at least K(M+1) each and k K(M+1) ≥ W_B for k ≥ 2.

- Outcome-based motives with votes and vote shares contractible:
  - Proposition 7 (constructive contract): For v^−1(K(M+ 1)) ≤ i ≤ M+ 1, transfers specified as:
    - ti =
      - 0 if ci = a and #a ≥ M+ 1
      - K(M+ 1) − vi if ci = a and #a < M+ 1
      - 0 if ci = b
    - For i < v^−1(K(M+ 1)) or i > M+ 1: no contract offered.
  - Cost of this contract is zero in equilibrium; deviations by voters are unprofitable; B cannot profitably recruit k ≥ 2 voters because each would require at least K(M+1) and k K(M+1) ≥ W_B.

### Comparative findings and mechanism insights
- Matching contractibility to voter motives minimizes cost:
  - Vote-contingent contracts are most cost-effective when voters have vote-based motives.
  - Outcome-contingent contracts are most cost-effective when voters have outcome-based motives.
- Misalignment between what is contractible and what voters care about:
  - Raises the cost of successful vote buying and can make it impossible (Proposition 3 example).
- Enriching contract space (votes + vote shares):
  - Greatly increases interest groups' ability to buy votes cheaply; unpopular policies can be secured at arbitrarily small cost despite large opposition.
  - Competition between interest groups offers little protection when contract space is rich.
- Secret ballot implications:
  - Under vote-based motives, secret ballots are effective deterrents.
  - Under outcome-based motives, secret ballots are much less effective.
- Policy prescriptions:
  - Make contingent contracts extremely costly to prevent cheap vote buying (e.g., heavy fines, forfeiture of office).
  - Use a combination of measures: secret ballots, enfranchisement, enforcement.
- Information technology and transparency:
  - Greater transparency and lower contracting costs can make modest expenditures effective at altering outcomes.

### Key numeric and formulaic items (verbatim)
- C^0_A = Σ_{i=1}^M max{0, −v_i}
- C_B(K, m) = (m − M + 1) K
- K(m) = W_B / (m − M + 1)
- v^−1(K(m)) = lowest index i such that v_i < K(m)
- C_A(m) = Σ_{i=v^−1(K(m))}^m t_i
- For outcome-based deterrence:
  - ϕ(m) = W_B / (m − M + 2) = K(m+1)
- Cost under outcome-based candidate contract:
  - C_A(m) = z(m) K(m+1) + Σ_{i=z(m)+1}^m (K(m) − v_i)
- CA(m) = sum_{i=1}^m max ( -vi ; 0 )
- Least-cost successful contract costs A an amount C0A - v(M+ 1)
- Payment rules for Propositions 6 and 7 (verbatim structure):
  - For v^−1(K(M+ 1)) ≤ i ≤ M+ 1
    - ti =
      - max ( -vi ; 0 ) if ci = a and #a ≥ M+ 1
      - K(M+ 1) − vi if ci = a and #a < M+ 1
      - 0 if ci = b
    - For i < v^−1(K(M+ 1)) or i > M+ 1; no contract is offered.
- Deterrence condition used repeatedly:
  - To induce k ≥ 2 voters, each must be paid at least K(M+ 1) and k K(M+ 1) ≥ W_B for k ≥ 2.

*Source: _wp0611 - 1. Therefore, aK(m=M)contract is the least-cost way to achieve a bare majority.*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   20

### _wp0611 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   20

### I. Introduction — context and examples of vote buying
- Historical and contemporary instances of vote buying:
  - George Washington (1757) offered constituents "an average of one and a half quarts of various alcoholic beverages" in exchange for votes.
  - Pasuk et al. (2000) survey evidence: in the 1996 Thai general elections, "fully 30 percent of heads of households were offered money in exchange for their votes."
  - 19th-century Maryland practice of negative vote buying: offering payment in exchange for not voting.
  - Post-1872 England: offering villages or neighborhoods material incentives contingent on aggregate turnout after the introduction of the secret ballot.
  - 2000 Taiwan Presidential election: the ruling National Party subsidized betting parlors to offer extremely favorable betting odds on the event that the party's candidate was elected.
  - 1830s English boroughs: price of a vote "low early on election day and to increase as the day progressed," falling once a sufficient margin of victory was judged.
  - International Olympic Committee (IOC) scandal: key members reportedly paid an up-front amount for their individual vote plus a "bonus" conditional on the outcome of the vote (reported in BBC News, December 12, 1998).
- Types of vote-buying contracts highlighted:
  - Contracts specifying payment in exchange for an individual vote.
  - Negative vote buying (payment for not voting).
  - Contracts contingent on aggregate outcomes (e.g., turnout or election outcome).
  - Contracts with payments contingent on vote share or timing (e.g., increasing price during election day).
  - Indirect schemes (e.g., subsidizing third parties like betting parlors).

### Model focus and comparative contexts
- Central research question:
  - How does variation in the type of contracts available for vote buying affect the "buyability" of voters when interest groups compete?
- Key modeling insight:
  - The interaction between (a) what is contractible and (b) what voters care about determines the success and cost of vote buying.
- Context distinctions affecting voter preferences:
  - Legislator voters: primarily concerned with how their voting record is perceived by constituents (individual-vote concern).
  - Citizen voters in large general elections: primarily concerned with the outcome (policy outcome concern).

### Main results (enumerated findings)
- 1. "Vote buying is cheaper when what is contractible coincides with what voters care about."
  - Example: Contracts contingent on an individual's vote are more effective when voters mainly care about their own vote (typical for legislators accountable to constituents).
  - Example: Contracts contingent on policy outcomes are more effective when voters mainly care about policy outcomes (typical in large general elections).
- 2. "Vote buying can become extremely costly, or even impossible, when what is contractible does not coincide with what voters care about."
  - Example: When voters care about their individual votes but only outcomes are contractible, there is no "price" at which an interest group can guarantee its preferred outcome through vote buying.
- 3. "Vote buying becomes extremely cheap, or even free, when both individual votes and vote shares are contractible."
  - Competition among interest groups has little effect in counteracting this outcome.
  - The authors construct a contract that guarantees a supermajority of votes to an interest group at a cost equal to or only slightly above the level that would obtain in the absence of competition.

### Structure of paper (forward roadmap)
- The paper proceeds with the presentation of the basic model in Section II.

*Source: _wp0611 - References . . . . . . . . . . . . . . . . . . . . . . . . . .   20*

### Section III examines equilibrium outcomes of vote buying under three contractual

### _wp0611 - Section III examines equilibrium outcomes of vote buying under three contractual

### II. The Model
- Environment:
  - An odd number, n, of voters choose between two policies labeled a (the new policy) and b (the status quo). The policy receiving the majority of votes is adopted.
  - Two interest groups, A and B, prefer a and b respectively. Excluding vote buying costs, group A enjoys payoff W_A > 0 when a is adopted and zero when b is adopted; group B enjoys payoff W_B > 0 when b is adopted and zero when a is adopted.
  - Voters i = 1,2, …, n care about their actual vote c_i, the policy outcome p, and transfers t_i. Voter utility: U_i(c_i, p, t_i) = u_i(c_i, p) + t_i.
  - Two polar specifications for u_i:
    - Vote-based motives: u_i independent of p. The difference from changing own vote from b to a is v_i = u_i(c_i = a) − u_i(c_i = b).
    - Outcome-based motives: u_i independent of c_i. The difference from outcome b to a is v_i = u_i(p = a) − u_i(p = b).
  - v_i are common knowledge, v_i ≠ 0 for all i, and v_i is strictly decreasing in index i. The median voter M = (n+1)/2 prefers policy b: v_M < 0. Without A, policy b would be adopted.
  - Game timing (extensive form): A proposes a contract; after observing A's proposal, B proposes a contract; then votes are cast and payoffs realized. If B is indifferent between proposing and staying out, B stays out.

- Definitions and equilibrium selection:
  - Definition 1: A vote buying contract is successful iff it guarantees adoption of policy a (i.e., all subgame perfect equilibria following the contract lead to adoption of a).

### III. Variation in Contracts — Overview
- The paper compares equilibrium outcomes of vote buying under three contractual variations:
  - Contracts contingent only on votes.
  - Contracts contingent only on outcomes.
  - Contracts contingent on votes and vote shares.
- The analysis starts with the benchmark identical to Groseclose and Snyder (GS): individual votes contractible and voters have vote-based motives.
- Multiplicity of equilibria can arise when contracts create interactive incentives (e.g., contingencies on vote shares or outcomes) or when voters have outcome-based motives. The analysis adopts a conservative definition of success (Definition 1) to address multiplicity.

### III.A Contracting on Votes
- Vote-Based Motives (benchmark, GS case):
  - Absent competition (A as monopoly), A can offer transfers t_i = −v_i to all voters i ≤ M whose intrinsic preferences favor b (v_i < 0). No transfers needed for voters who prefer a.
  - Monopoly vote buying cost:
    - C^0_A = Σ_{i=1}^M max{0, −v_i}
  - With competition, fix coalition size m with n ≥ m ≥ M. Define K as the minimum expected payoff earned by any voter i ∈ {1,…,m} (including transfers from A). Choose K so that B will (just) not wish to invade A's coalition. For B to obtain its desired policy it must re-bribe at least m − M + 1 voters, so
    - C_B(K, m) = (m − M + 1) K
    - To deter B, require C_B(K, m) ≥ W_B. Solving equality yields:
      - K(m) = W_B / (m − M + 1)
  - Define v^−1(K(m)) to be the lowest index i such that v_i < K(m). A's proposed K(m)-contract (for coalition size m):
    - For v^−1(K(m)) ≤ i ≤ m:
      - t_i = K(m) − v_i if c_i = a
      - t_i = 0 if c_i = b
    - For i < v^−1(K(m)) or i > m: no contract offered.
  - Cost of this scheme:
    - C_A(m) = Σ_{i=v^−1(K(m))}^m t_i
  - A chooses m to minimize C_A(m); let m^* ∈ arg min_m C_A(m) and, if multiple minima, choose the largest integer in arg min set.
  - A participates only if C_A(m^*) < W_A.
  - Proposition 1: A K(m^*) contract is a least-cost successful contract under vote-based motives where only individual votes are contractible.
  - Cost characterization: C_A(m^*) can be rewritten as
    - C_A(m^*) = C^M_A + W_B + { (M − v^−1(K(m^*))) / (m^* − M + 1) W_B − Σ_{i=v^−1(K(m^*))}^{v^−1(0)−1} v_i } + Σ_{i=M+1}^{m^*} (−v_i)
    - The expression in curly brackets is positive; hence cost to A of vote buying is strictly greater than its monopoly cost plus the entire value to group B. Thus A must have a high valuation W_A to undertake vote buying if it can only condition on votes.
  - Note: If A could contract with B, A could pay B amount W_B to stay out and obtain a at cost C^M_A, yielding total cost W_B + C^M_A < C_A(m^*). The analysis excludes this possibility following GS.

- Outcome-Based Motives (when only votes are contractible):
  - Intuition conflicts: voters who care only about outcomes might be easier to influence off the pivot but contracts are only on votes, so contracting may be more difficult.
  - If a voter believes he is pivotal, margins are identical to vote-based case; offering a K(m)-contract can deter B from recruiting a bare-majority coalition.
  - If a voter believes he is not pivotal (B attempts to recruit a supermajority), then v_i acts like zero for that voter; B's cost is increasing in supermajority size, so A must defend against coalition of size M+1 as well.
  - For the alternate problem with v_i = 0 for all i, to deter B from recruiting M+1, A must offer m − M voters transfers of at least
    - ϕ(m) = W_B / (m − M + 2) = K(m+1)
    - Hence B must pay at least (m − M + 2) ϕ(m) = W_B to recruit required extra voters.
  - Candidate least-cost successful contract recruiting m voters:
    - For i ≤ m:
      - t_i = max{K(m+1), K(m) − v_i} if c_i = a
      - t_i = 0 if c_i = b
    - For i > m: no contract offered.
  - Define z(m) to be the largest integer such that
    - K(m+1) ≥ K(m) − v_{z(m)}
    - Cost to A:
      - C_A(m) = z(m) K(m+1) + Σ_{i=z(m)+1}^m (K(m) − v_i)
    - Let m^^ be an integer m ≤ n that minimizes A's cost; if multiple minimizers, choose the largest.
  - Proposition 2: The contract with m = m^^ described above is a least-cost successful contract under outcome-based motives where only individual votes are contractible. Furthermore, this contract is more costly than the least-cost successful contract under vote-based motives where only individual votes are contractible.
  - Proof sketch highlights:
    - When A recruits a supermajority, recruited voters ascribe infinitesimal probability of being pivotal and vote for the side offering higher bribe (A). Unbribed voters face small pivot probability and vote for a iff v_i > 0. B cannot profitably overturn because either recruiting a bare majority costs at least W_B or constructing the required supermajority costs at least W_B by definition of K(m^^ + 1).
  - Coalition size comparisons:
    - No unambiguous ranking of m^* (vote-based optimal coalition) and m^^ (outcome-based optimal coalition). Example with linear v_i = β − γ(i − M):
      - With γ = 1, β = (n+1)[1 − p/2 4 p^2] (as specified), and W_B = 1/2 (M)^2:
        - When n = 9: m^* = 8 and m^^ = 7.
        - When n = 99: m^* = 84 and m^^ = 86.
    - Marginal cost decomposition (ignoring integer constraint) when m = m^*:
      - ∂C_A(m)/∂m |_{m=m^*} = K'(m^* + 1) z(K(m^*)) + (v^−1(K(m^*)) − z(K(m^*))) K'(m^*)
      - First term: savings defending against a supermajority when recruiting larger coalition. Second term: loss defending against a bare majority when recruiting larger coalition. Either effect can dominate.
    - Remark 1: In large elections where intrinsic preferences are approximately linear, larger fractions of the electorate will be bribed under outcome-based motives than under vote-based motives. Formally, if v_i = β − γ(i − M), then for n sufficiently large, m^^ > m^*.

- Empirical note:
  - The least-cost successful contract shares features with documented vote buying strategies (Quimbo, 2002): bribe amounts may vary among supporters, undecided, and opposition; undecided sometimes get three times as much as supporters; key opposition supporters receive even more to switch.

- Discussion (comparing Propositions 1 and 2):
  - When voters care about outcomes, successful contracts require paying even staunch supporters of a for their votes. Ubiquity of vote buying in this case aligns with anecdotal evidence from general elections (Pasuk et al., 2000; Shaffer, 2002).

### III.B Contracting on Outcomes
- Motivation: Secret ballots prompted parties to make payments conditional on outcomes or turnout at the village/neighborhood level (historical evidence cited).
- Setting: Interest groups can only offer contracts contingent on the policy outcome (a or b), not on individual votes. Transfers conditional on the unfavored policy are never optimal.
- Vote-Based Motives when only outcomes are contractible:
  - Intuition that vote buying is easier is wrong because voters care about their own voting record (which is not contractible). Opacity plus career concerns reduces vote buying.
  - Impossibility result:
    - Proposition 3: If v_{M−1} < 0, then successful vote buying contracts do not exist under vote-based motives where only outcomes are contractible.
    - Proof sketch: If b commands a supermajority absent vote buying, each voter has only infinitesimal probability of being pivotal; changing vote causes first-order payoff v_i, so voters vote according to intrinsic preferences and B need not incur cost, so A's outcome-contingent contract has vanishingly small incentive effect.
  - When intrinsic support for b is only a bare majority, successful vote buying is possible but buying a supermajority is impossible when only outcomes are contractible. Thus the cost to A of successful vote buying is increased in this contractual restriction.
  - Proposition 4: If v_{M−1} > 0, then the K(M) contract is a least-cost successful contract under vote-based motives where only outcomes are contractible.
    - Argument highlights:
      - If a voter ascribes infinitesimal probability of being pivotal, she votes for a iff v_i > 0. If a voter ascribes probability close to one of being pivotal, she votes for a iff v_i + t_i > b (where b is bribe offered by B conditional on outcome b).
      - Any successful contract by A must induce a bare majority (not a supermajority), because if a supermajority is induced then no voter is pivotal and voters would revert to intrinsic preferences, contradicting success.
      - The K(M) contract induces a bare majority and, when a bare majority votes for a, payoffs under the K(M) scheme equal those when contracts are contingent on individual votes (as in Proposition 1). Thus K(M) is least-cost successful in this case.

*Source: _wp0611 - Section III examines equilibrium outcomes of vote buying under three contractual*

### 1. Therefore, aK(m=M)contract is the least-cost way to achieve a bare majority.

### _wp0611 - 1. Therefore, aK(m=M)contract is the least-cost way to achieve a bare majority.

### Main propositions and results
- Proposition 1 / baseline result (implicit): a K(m = M) contract is the least-cost way to achieve a bare majority. If B does nothing, this contract is successful. B cannot do better than offering no contract.
- Remark 2: A K(M) contract is a least-cost contract such that there exists a subgame perfect equilibrium in which policy a is adopted.
- Proposition 5 (Outcome-Based motives): The K(m
  
  ) contract is a least-cost successful contract under outcome-based motives where only outcomes are contractible. Furthermore, the costs of this scheme are equal to the costs of a least-cost successful contract under vote-based motives where only votes are contractible.
  - Proof sketch: The voter's payoff difference between voting for a versus b hinges entirely on the pivotal case; conditional on being pivotal payoffs are identical to vote-based motives where contracts condition on individual votes. Argument identical to Proposition 1 yields the result.

### Contracting on votes and vote shares — key constructions
- Notation: #a denotes the number of votes cast for policy a. Number of votes for b is n - #a.
- Vote-based motives (competition with vote-share contingencies):
  - When individual votes and vote shares are contractible, competition has virtually no effect on A's minimum cost: least-cost successful contract costs A an amount C0A - v(M+ 1).
  - Proposition 6 (least-cost successful contract under vote-based motives where individual votes and vote shares are contractible):
    - For v
      1
      (K(M+ 1)) ≤ i ≤ M+ 1
      ti =
        - max ( -vi ; 0 ) if ci = a and #a ≥ M+ 1
        - K(M+ 1) - vi if ci = a and #a < M+ 1
        - 0 if ci = b
    - For i < v
      1
      (K(M+ 1)) or i > M+ 1; no contract is offered.
    - Proof highlights:
      - Minimum cost of obtaining #a = m votes is CA(m) = sum_{i=1}^m max(-vi ; 0).
      - A contract with #a = M would allow B to capture one voter at arbitrarily small cost, so it fails to implement a.
      - Under the proposed contract, if A offers it, B is deterred because to recruit k ≥ 2 voters it must pay each at least K(M+1) and k K(M+1) ≥ WB for k ≥ 2.
  - Intuition: A compensates "loyal" voters when a's success is in doubt and pays minimally when a has a supermajority; this entry-deterrence strategy raises B's invasion costs while keeping A's equilibrium payments low.

- Outcome-based motives with votes and vote shares contractible:
  - Proposition 7 (least-cost successful contract under outcome-based motives where individual votes and vote shares are contractible):
    - For voters v
      1
      (K(M+ 1)) ≤ i ≤ M+ 1
      ti =
        - 0 if ci = a and #a ≥ M+ 1
        - K(M+ 1) - vi if ci = a and #a < M+ 1
        - 0 if ci = b
    - For i < v
      1
      (K(M+ 1)) or i > M+ 1; no contract is offered.
    - Proof highlights:
      - Cost of the contract is zero in equilibrium, hence least-cost.
      - Deviations by voters are unprofitable because expected payoff differences (functions of probabilities 1, 2, 3) are negative.
      - B cannot profitably recruit k ≥ 2 voters because it would have to pay at least K(M+1) to each and k K(M+1) ≥ WB.

### Comparative findings and mechanism insights
- Match between contractual contingencies and voter motives minimizes cost:
  - Vote-contingent contracts are most cost-effective when voters have vote-based motives.
  - Outcome-contingent contracts are most cost-effective when voters have outcome-based motives.
  - Misalignment of contractible contingencies and voter preferences raises the cost of successful vote buying or can make it impossible.
- Enriching contract space (conditioning on individual votes and aggregate vote share) greatly increases the ability of interest groups to buy votes cheaply; in extreme cases an unpopular policy can be secured at arbitrarily small cost despite a supermajority of electorate opposition.
- Secret ballot:
  - Under vote-based motives (voters care mainly about their individual votes), secret ballots are a powerful remedy against vote buying.
  - Under outcome-based motives (voters care mainly about policy outcomes), secret ballots do much less to raise the price of successful vote buying.
- Policy deterrence via contingent-contract costs:
  - Contracts that condition on votes and vote shares allow A to promise high compensation when its proposal is at risk, deterring B without large equilibrium payouts by A.
  - Policy prescription: contingent contracts of the specified form must be made extremely costly to prevent cheap vote buying, potentially via penalties such as forfeiture of office or heavy fines.

### Discussion and broader implications
- Competition between interest groups:
  - When contract space is rich (votes and vote shares contractible), competition offers little protection; interest groups can still obtain preferred outcomes at arbitrarily small cost.
  - Mere size of the electorate does not deter vote buying in these contractual environments because total expenditure need not increase much with the number of voters needed to win.
- Institutional remedies should be combined:
  - No single measure prevents vote buying in all circumstances.
  - Successful policy requires a combination of secret ballots, enfranchisement, and strategic enforcement.
- Information technology and transparency:
  - Increased transparency combined with lower communication and contracting costs can make modest vote-buying expenditures highly effective at altering policy outcomes.

### Key numeric and formulaic items (preserved verbatim)
- CA(m) = sum_{i=1}^m max ( -vi ; 0 )
- Least-cost successful contract costs A an amount C0A - v(M+ 1)
- Payment rules for Propositions 6 and 7 (verbatim):
  - For v
    1
    (K(M+ 1)) ≤ i ≤ M+ 1
    ti =
      - max ( -vi ; 0 ) if ci = a and #a ≥ M+ 1
      - K(M+ 1) - vi if ci = a and #a < M+ 1
      - 0 if ci = b
    - For i < v
      1
      (K(M+ 1)) or i > M+ 1; no contract is offered.
  - (Same structure for outcome-based motives with the zero/nonzero pattern stated in Proposition 7.)
- Conditions used in deterrence proofs:
  - To induce k ≥ 2 voters, each must be paid at least K(M+ 1) and k K(M+ 1) ≥ WB for k ≥ 2.

*Italicized source attribution: _wp0611 - 1. Therefore, aK(m=M)contract is the least-cost way to achieve a bare majority.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp0611.pdf_