Does Majority Rule Produce Hasty Decisions?

Transcription

1 Does Majority Rule Produce Hasty Decisions? Jimmy Chan 1 Wing Suen 2 1 Fudan University 2 University of Hong Kong October 18, 2013 Ohio State University Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 1 / 29

2 Introduction Madison, The Federalist Papers, No. 58: [The supermajority requirement] might have been an additional shield to some particular interests, and another obstacle generally to hasty and partial measures. [A]n interested minority might take advantage of it to screen themselves from equitable sacrifices to the general weal, or in particular emergencies to extort unreasonable indulgences. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 2 / 29

3 Introduction Collective decision-making is not just about casting a vote. A standards setting committee involves engineers to test and discuss the proposed standards. Countries deciding on whether and how to cut greenhouse gas emissions engage a panel of scientists to gather evidence to guide their actions. During jury deliberation, the jurors together examine and learn from the evidence presented at trial to form an opinion for a verdict. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 3 / 29

4 Introduction These examples share common features: Information discovered at the deliberative stage is public knowledge to the group members. Information discovery is costly Members of the group themselves can decide to keep deliberating until they reach a decision. Members have different preferences for alternatives as well as different trade-offs between quality of decisions versus speed patient vs. impatient agents high-stake vs. low-stake agents Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 4 / 29

5 Introduction Use a collective version of the sequential probability ratio test (Abraham Wald 1947) to study the trade-off between the cost of collecting additional information and the benefit from making a more informed decision. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 5 / 29

6 Related Literature Collective search/experimentation: Albrecht, Anderson and Vroman (2010); Compte and Jehiel (2010a; 2010b ); Strulovici (2010): loss of control Lizzeri and Yariv (2012): one-dimensional heterogeneity Gul and Pesendorfer (2012): two political parties Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 6 / 29

7 Model 2m 1 agents are choosing between two alternatives: α and β. In state A, agent i s payoff to α is 1, and his payoff to β is 0. In state B, agent i s payoff to α is 0, and his payoff to β is e v i. θ 0 = log(pr[ω = A]/ Pr[ω = B]) represents the common initial belief. The immediate expected payoff from α is higher than β if and only if θ v i. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 7 / 29

8 Model At each time t, each agent independently votes for α or β, or abstains. Decision rule is k {m, m + 1,..., 2m 1}: an alternative is adopted at time t if it is supported by k agents or more; the decision process continues if neither α nor β receives sufficient votes. k = m corresponds to majority rule. Agent i discounts future payoff at a rate r i. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 8 / 29

9 Model Information accumulation is represented by a Wiener process ds: State A: ds = µdt + ρdw State B: ds = µdt + ρdw Accumulated evidence S t is a sufficient statistic. Log-likelihood ratio of observing S t = s is 2µs/ρ 2. Let S t = 2µS t/ρ 2. Common belief at time t is: θ t = θ 0 + S t. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 9 / 29

10 Model Focus on Markov threshold strategies: votes for α when θ t G i, votes for β when θ t g i, and abstains when θ t (g i, G i ). Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 10 / 29

11 Model Let G be the k-th smallest G i among the group. if θ t > G, α is adopted payoff to agent i is e θt /(1 + e θt ) Let g be the k-th largest g i among the group. if θ t < g, β is adopted payoff to agent i is e v i /(1 + e θ t ) If θ t [g, G], deliberation continues. Agent i s payoff satisfies: u i (g, G θ) = e r i dt E[u i (g, G θ + ds )] Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 11 / 29

12 Strategic Best Response in Stopping Decisions Φ i (g) is the optimal G that maximizes u i (g, G θ) subject to G g. It is the one-sided best-response stopping boundary for taking decision α if the lower boundary for taking β is fixed at g. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 12 / 29

13 Strategic Best Response in Stopping Decisions (g i,g i ) G (v i,v i ) φ i Φ i g Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 13 / 29

14 Strategic Best Response in Stopping Decisions G v i φ i (.; r i ) φ i (.; r i ) φ i (.; r i ) r i > r i > r i g Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 14 / 29

15 Strategic Best Response in Stopping Decisions Waiting has no value if g v i. Waiting has little value if g is near v i. Non-monotonicity reflects loss of control. The best-response waiting window can be arbitrarily narrow if lower boundary is fixed at near v i or if r i is very high. Φ i (g) G i for all g v i. There is a strategic difference between too much waiting and too little waiting. Φ i (g) can be lower than v i : agent prefers to stop and take α even though β is better than α at that point. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 15 / 29

16 Equilibrium Analysis Pivotal best response functions: Φ piv (g, k) is the k-th smallest Φ i (g) for each g. (ĝ, Ĝ) is an equilibrium if and only if it is a fixed point of (Φ piv, φ piv ). One-sided best-response suffices for the two-sided problem. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 16 / 29

17 Equilibrium Analysis G φ 2 φ 3 φ 1 G v 3 G v 2 v 1 φ piv Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 17 / 29

18 Equilibrium Analysis Equilibrium Ĝ is no greater than the k-th smallest G i. If all agents have the same discount rate, then equilibrium is unique. If agents have different discount rates, multiple equilibria may exist. In the most patient and the least patient equilibrium, Ĝ ĝ decreases in r i. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 18 / 29

19 Rushing to a Decision One vote is sufficient to tilt the decision from α to β (or vice versa) under majority rule k = m. When r i is sufficiently high, agent i will always be pivotal against agent m under majority rule. This impatient agent i is willing to wait very little before voting for an alternative. Strategic complementarity in the decision to stop early then suggests that the other pivotal agent m will respond by cutting short his stopping threshold as well. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 19 / 29

20 Rushing to a Decision Called a hasty equilibrium, because, starting at any initial belief θ 0 (ĝ, Ĝ), the time it takes for the belief θ t = θ 0 + S t to reach the boundaries is short. Whether the true state is A or B has little effect on the probability of choosing α. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 20 / 29

21 Rushing to a Decision (a) r 1 = r 2 = r 3 (b) r 1 > r 2 = r 3 (c) r 1 >> r 2 = r 3 φ 2 φ 3 φ 2 φ 3 φ 2 φ 3 G Φ 3 Φ 2.P1 v 2 v 3 G Φ 3 Φ 2 P2. v 2 v 3 G Φ 3 Φ 2 P3. v 2 v 3 v 1 v 1 v 1 φ 1 Φ 1 g g g Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 21 / 29

22 Rushing to a Decision Think of an impatient agent as a swing voter. He may be biased in favor of α, but because of discounting he is willing to settle for β as long as the evidence swings a bit to the left. These agents are easy targets of alliance. To avoid some members of the group from capturing these impatient voters to adopt one alternative, the other members who are biased for the other alternative will cut short the deliberative process by pushing forward the stopping threshold for their favored alternative as well. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 22 / 29

23 Rushing to a Decision The effects of very patient agents and very impatient agents are not symmetric. patient agents are less likely to be pivotal patient agents induce strategic substitution by other agents Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 23 / 29

24 Rushing to a Decision φ 1 φ 3 Φ 3 P1. (a) r 2 = r 1 = r 3 Φ 3. P4 (b) r 2 << r 1 = r 3 φ 1 φ 3 Φ 1 v 3 Φ 1 v 3 G v 2 G v 2 v 1 v 1 φ 2 Φ 2 g g Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 24 / 29

25 Supermajority Rule is More Robust Ĝ ĝ increases in k. We can avoid a hasty equilibrium using decision rule k if the number of impatient agents does not exceed 2(k m). There are at least 2k (2m 1) = 2(k m) + 1 agents who are in the winning coalition for both α and β. At least one agent who is in the winning coalition for both α and β is not impatient. Such a patient agent cannot be maximizing his utility in a hasty equilibrium. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 25 / 29

26 Supermajority Rule is More Robust If the number of very patient agents does not exceed 2m k 1, we can avoid an equilibrium with excessively long deliberations. Example: 1/4 very impatient agents, 1/4 very patient agents, 1/2 normal agents. A super-majority rule with between 5/8 to 3/4 super-majority requirement can avoid very short or very long deliberations. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 26 / 29

27 Supermajority Rule Has its Drawbacks Majority rule (k = m) always respects the static preferences of agents: the alternative chosen is preferred by a majority of the group. Supermajority rule (k > m) does not have this property: there exist preference profiles such that in equilibrium only 2m-k agents favor the alternative chosen by the group. Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 27 / 29

28 Supermajority Rule Has its Drawbacks Supermajority rule can produce an extortive equilibrium: A minority can hold out for its favored decision, with the remaining majority agreeing to it simply because they don t want to wait too long. Φ i (ĝ) falls without bound as ĝ goes to minus infinity this is an extreme manifestation of strategic substitution supermajority rule lengthens deliberation but does not necessarily promote consensus Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 28 / 29

29 Thank you! Chan/Suen (Fudan/HKU) Majority Rule and Hasty Decisions 29 / 29