Online Appendix for “Gambler’s Fallacy and Imperfect Best Response in Legislative Bargaining” Salvatore Nunnari Bocconi University and IGIER
[email protected] Jan Z´apal CERGE-EI & IAE-CSIC and Barcelona GSE
[email protected] June 21, 2016
Contents 1 Numerical Computations of QRE and QGF: Additional Results
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2 Empirical vs. Estimated Distributions
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3 Systematic Search for QRE Multiplicity
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4 Optimal QRE Proposals
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5 QRE Expected Payoffs
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6 QRE & QGF Estimation with Alternative Datasets
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7 Log-likelihood of SSPE benchmark
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8 Proposals Distribution and Acceptance Probabilities in QRE
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9 Proposals Distribution in Experimental Data
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10 Stationary Version of Gambler’s Fallacy Model
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This document contains supplementary material for ‘Gambler’s Fallacy and Imperfect Best Response in Legislative Bargaining’.
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1
Numerical Computations of QRE and QGF: Additional Results
All figures with n = 3 and φ = 1 (for QGF). Unless otherwise specified, the first row of the following figures (panels (a), (b)) shows the QRE, the second row (panels (c), (d)) shows the QGF and the third row (panels (e), (f)) shows both, for comparison. The first column of the figures (panels (a), (c), (e)) shows results for δ = 1 and the second column (panels (b), (d), (f)) shows results for δ = 12 .
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Figure 1: QRE vs. QGF in Baron and Ferejohn (1989) model Mean proposer’s share (a)
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Figure 2: QRE vs. QGF in Baron and Ferejohn (1989) model Mean approved proposer’s share (a)
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Figure 3: QRE vs. QGF in Baron and Ferejohn (1989) model Mean proposer’s share conditional on MWC (a)
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Figure 4: QRE vs. QGF in Baron and Ferejohn (1989) model Mean approved proposer’s share conditional on MWC (a)
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Figure 5: QRE vs. QGF in Baron and Ferejohn (1989) model Equilibrium continuation value v ∗ (a)
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Figure 6: QRE vs. QGF in Baron and Ferejohn (1989) model MWC frequency (a)
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Figure 7: QRE vs. QGF in Baron and Ferejohn (1989) model Expected number of rounds (a)
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Figure 8: QRE vs. QGF in Baron and Ferejohn (1989) model Equilibrium continuation value in QGF (a)
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Note: Equilibrium odd round in QGF. First row of proposing player. Second row v ∗ of non-proposing player. Left column δ = 1. Right column δ = 12 .
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2
Empirical vs. Estimated Distributions
Each figure of this section shows the distribution of the proposer’s share and the probability of approving the proposed allocation derived from the experimental data (round 1 observations only) and predicted by QRE and QGF at ˆ and {λ, ˆ φ} ˆ respectively. the benchmark MLE λ
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Figure 9: Distribution of proposer’s share and probability of approving vote Experimental data vs. QRE vs. QGF 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%
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Note: Experiment 1 (Frechette, Kagel, and Lehrer, 2003). Dashed line QRE with ˆ = 20.2. Dotted line QGF with λ ˆ = 21.7 and φˆ = 1. Vertical lines benchmark SSPE λ ∗ predictions (v and proposer’s share).
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Figure 10: Distribution of proposer’s share and probability of approving vote Experimental data vs. QRE vs. QGF 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%
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Note: Experiment 2 (Frechette, Kagel, and Morelli, 2005b). Dashed line QRE with ˆ = 22.1. Dotted line QGF with λ ˆ = 21.7 and φˆ = 3. Vertical lines benchmark SSPE λ predictions (v ∗ and proposer’s share).
Figure 11: Distribution of proposer’s share and probability of approving vote Experimental data vs. QRE vs. QGF 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%
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Note: Experiment 3 (Frechette, Kagel, and Morelli, 2005b). Dashed line QRE with ˆ = 23.4. Dotted line QGF with λ ˆ = 22.5 and φˆ = 1. Vertical lines benchmark SSPE λ ∗ predictions (v and proposer’s share).
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Figure 12: Distribution of proposer’s share and probability of approving vote Experimental data vs. QRE vs. QGF 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%
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Note: Experiment 4 (Frechette, Kagel, and Morelli, 2005b). Dashed line QRE with ˆ = 10.6. Dotted line QGF with λ ˆ = 13.3 and φˆ = 1. Vertical lines benchmark SSPE λ predictions (v ∗ and proposer’s share).
Figure 13: Distribution of proposer’s share and probability of approving vote Experimental data vs. QRE vs. QGF 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%
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Note: Experiment 5 (Frechette, Kagel, and Morelli, 2005a). Dashed line QRE with ˆ = 33.5. Dotted line QGF with λ ˆ = 33.8 and φˆ = 1. Vertical lines benchmark SSPE λ ∗ predictions (v and proposer’s share).
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Figure 14: Distribution of proposer’s share and probability of approving vote Experimental data vs. QRE vs. QGF 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%
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Note: Experiment 6 (Drouvelis, Montero, and Sefton, 2010). Dashed line QRE with ˆ = 18.4. Dotted line QGF with λ ˆ = 18.8 and φˆ = 1. Vertical lines benchmark SSPE λ predictions (v ∗ and proposer’s share).
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3
Systematic Search for QRE Multiplicity
The figures show the systematic search for multiplicity of QRE. Each figure plots σ λ (v) for v ∈ [0, 1] and λ ∈ {0, 2, 6, 10, 18, 36, 72, 144} for the model with n = 3.
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Figure 15: QRE σ(v) mapping (a) δ = 1/6
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Optimal QRE Proposals
The following figures characterize optimal QRE proposals for λ ∈ [0, 200] and all possible combinations of n ∈ {3, 5} and δ ∈ { 12 , 1}. Optimal proposal x∗ ∈ X 0 maximizes xi pλv∗ (x) + δv ∗ (1 − pλv∗ (x)) where v ∗ is the equilibrium continuation value and pλv∗ (x) is the equilibrium probability of proposal x being accepted. x∗ can be characterized by two numbers: x∗i the proposer allocates to herself (top figure of each panel) and n∗ , the number of coalition partners with strictly positive shares in x∗ (bottom figure of each panel).
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Figure 16: QRE in Baron and Ferejohn (1989) . . . Optimal proposal (a) n = 3 & δ = 1 1
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Figure 17: QRE in Baron and Ferejohn (1989) . . . Optimal proposal (a) n = 5 & δ = 1 1
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Note: Top figure of each panel is proposer’s share (thick line) and responder’s share (thin line) in proposer’s optimal proposal. Bottom figure of each panel is # of coalition partners with strictly positive shares in optimal proposal.
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5
QRE Expected Payoffs
The following figures show the proposer’s and responder’s expected payoffs in a QRE for n ∈ {3, 5} and δ ∈ { 12 , 1}. Denote the proposer by i. The proposer’s expected payoffs in a QRE with continuation value v ∗ is X rvλ∗ ,i (x) xi pλv∗ ,i (x) + δv ∗ (1 − pλv∗ ,i (x)) . x∈X 0
Denote by x(−i) the largest allocation in x after dropping xi . The responder’s expected payoff in a QRE with continuation value v ∗ is X rvλ∗ ,i (x) x(−i) pλv∗ ,i (x) + δv ∗ (1 − pλv∗ ,i (x)) . x∈X 0
The difference between the proposer’s and the responder’s expected payoff captures the proposer’s bargaining power. In the benchmark SSPE, the difference between the proposer’s and the responder’s expected payoff is n+1 δ δ n−1 δ . − =1− 1− 2 n n 2 n For n = 3 and δ ∈ { 12 , 1} this difference is, respectively, 23 and 31 . For n = 5 and 7 δ ∈ { 12 , 1} this difference is, respectively, 10 and 25 . These benchmark values can be read on the figures below. Since a non-proposing player is not certain to receive the largest allocation among the allocations the proposer distributes among the remaining players, denote the non-proposer’s expected payoff in a QRE with continuation value v ∗ by X rvλ∗ ,i (x) xj pλv∗ ,i (x) + δv ∗ (1 − pλv∗ ,i (x)) x∈X 0
where j 6= i. In the benchmark SSPE, the difference between the proposer’s and the non-proposer’s expected payoff is δ1 δ n−1 δ − =1− . 1− 2 n n2 2 For δ ∈ { 12 , 1} the difference is, respectively,
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Figure 18: QRE in Baron and Ferejohn (1989) . . . Expected proposer’s and responder’s payoff (a) n = 3 & δ = 1 1
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Figure 19: QRE in Baron and Ferejohn (1989) . . . Expected proposer’s and responder’s payoff (a) n = 5 & δ = 1 1
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Note: Proposer’s (solid line), responder’s (dashed line) and non-proposer’s (dotted line) expected payoff in QRE. The dash-dotted line sums one proposer’s and four non-proposer’s expected payoffs. The horizontal lines are the proposer’s and responder’s share in the benchmark SSPE.
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QRE & QGF Estimation with (In)Experienced Subjects, All Rounds and Paths of Play
ˆ for the QRE model and {λ, ˆ φ} ˆ for This section presents the MLE estimates of λ the QGF model using alternative datasets relative to the results included in the paper. Table 1 reports the benchmark estimates presented here for comparison. To reiterate, these estimates use all round 1 proposals. ˆ and {λ, ˆ φ} ˆ obtained using all round 1 proposTable 2 reports estimates λ als from experiments with inexperienced subjects (dropping periods 11-15 in experiment 1, 6-10 in experiment 6 and all data with re-invited subjects in experiments 2, 3, 4 and 5). Table 3 presents similar estimates but using all round 1 proposals from experiments with experienced subject. ˆ and {λ, ˆ φ} ˆ obtained using all proposals from all Table 4 shows estimates λ rounds. The inclusion of data from different rounds requires a slight alternation of the estimation strategy because a) QGF makes different predictions for odd and even rounds and b) both QRE and QGF predict different probability of proceeding beyond round 1 for different values of λ and φ. We deal with a) in a straightforward manner. Each observation of proposing or voting behavior comes with information about the round in which it was and we use round specific QGF predictions to calculate the maximum likelihood. For the variables in Table 4 that in the QGF model depend on odd and even rounds (average proposer’s share, quartiles, share of minimum winning coalitions) we present odd and even round specific predictions, weighted by the probability the game ends in odd and even rounds. Dealing with b) is slightly more complicated. For a given observation of proposing or voting behavior in round r > 1, we multiply the probability of that action being taken, as predicted by QRE or QGF, by the probability of the game proceeding to round r, also calculated using the QRE or QGF prediction. In other words, if the dataset contains observations from rounds r > 1, the MLE estimation penalizes high values of λ, which predict low rejection probabilities. Finally, Table 5 presents the estimates using the observed paths of play, i.e., selected proposals only. In each experiment and each period, we observe a series of proposed allocations x1 , . . . , xT containing T − 1 rejected proposals x1 , . . . , xT −1 and one accepted proposal xT . We call x1 , . . . , xT the path of play. For any λ in the QRE model or {λ, φ} in the QGF model we can calculate the likelihood of a given path of play as the probability of the event that all x1 , . . . , xT −1 are proposed and rejected and xT is proposed and accepted (using again odd and even round specific predictions in QGF). Summing over all the observed paths we obtain the likelihood of a given dataset, which we maximize ˆ or {λ, ˆ φ}. ˆ in order to estimate λ
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Table 1: Estimation results, round 1 data (benchmark included in the paper) Experiment
1
2
3
4
5
6
N δ Observations XP? R
5 4/5 275 .680
3 1 330 .666
3 1 411 .666
3 1/2 420 .833
5 1 450 .600
3 1 480 .666
.537 .500 .530 .600 .695 .093
.409 .350 .400 .500 .853 .422
.486 .420 .500 .570 .573 .269
10.6 .627 .540 .640 .730 .424 .179 -3432.5 -3346.2 -86.3
33.5 .400 .350 .400 .450 .773 .353 -3012.7 -2920.9 -91.8
18.4 .512 .460 .540 .600 .516 .443 -3849.9 -3662.4 -187.6
33.8 1 .387 .350 .400 .450 .794 .315 -2945.4 -2857.5 -87.9
18.8 1 .491 .440 .510 .570 .541 .382 -3766.0 -3560.9 -205.2
Data Avg(XP R ) Q1(XP R ) Q2(XP R ) Q3(XP R ) % MWC P[delay]
.338 .250 .350 .400 .422 .036
.553 .500 .530 .600 .727 .318
.522 .500 .500 .570 .793 .219 QRE
b λ bP R ) Avg(X bP R ) Q1(X bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
20.2 .441 .350 .450 .500 .547 .324 -2434.1 -2369.3 -64.8
22.1 .527 .490 .550 .600 .605 .345 -2380.4 -2300.1 -80.2
23.4 .532 .500 .550 .600 .635 .313 -2903.3 -2800.7 -102.7
QGF b λ φb bP R ) Avg(X b Q1(XP R ) bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
21.7 1 .421 .350 .450 .500 .553 .301 -2374.5 -2308.8 -65.7
21.7 3 .518 .480 .540 .590 .601 .339 -2377.7 -2296.4 -81.3
22.5 1 .497 .460 .510 .560 .637 .273 -2833.1 -2722.4 -110.6
13.3 1 .594 .530 .610 .680 .449 .152 -3309.2 -3237.2 -72.1
Likelihood Ratio Test (P-Values) Overall Proposing Voting
0.0000 0.0000 1.0000
0.0213 0.0066 1.0000
0.0000 0.0000 1.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0053
0.0000 0.0000 1.0000
Note: Experiment 1 from Frechette, Kagel, and Lehrer (2003); Experiments 2 through 4 from Frechette, Kagel, and Morelli (2005b); Experiment 5 from Frechette, Kagel, and Morelli (2005a); Experiment 6 ? , and X bP R refer, respectively to the proposer’s from Drouvelis, Montero, and Sefton (2010). XP R , XP R allocation observed in the data, the proposer’s allocation predicted by the benchmark model, and the proposer’s allocation predicted by the MLE estimates. % MWC refers to the incidence of minimum winning coalitions, defined as proposals where at least 1 member (for n = 3), or at least 2 members (for n = 5), receive less than 5% of the pie. All data and estimates refer to round 1 behavior.
25
Table 2: Estimation results, round 1 data, inexperienced subject only Experiment
1
2
3
4
5
6
N δ Observations XP? R
5 4/5 200 .680
3 1 240 .666
3 1 249 .666
3 1/2 300 .833
5 1 300 .600
3 1 240 .666
.517 .470 .500 .570 .630 .110
.405 .350 .400 .500 .807 .383
.465 .330 .500 .500 .417 .188
9.0 .617 .520 .640 .740 .410 .196 -2515.6 -2441.6 -74.0
32.3 .394 .350 .400 .450 .747 .382 -2097.2 -2041.0 -56.2
15.7 .503 .430 .530 .610 .457 .508 -2018.4 -1918.9 -99.5
32.4 1 .381 .350 .400 .450 .762 .347 -2060.3 -2006.0 -54.3
16.1 1 .486 .420 .510 .580 .473 .464 -1992.7 -1882.4 -110.3
Data Avg(XP R ) Q1(XP R ) Q2(XP R ) Q3(XP R ) % MWC P[delay]
.328 .250 .300 .400 .335 .050
.552 .500 .530 .600 .700 .350
.520 .500 .500 .600 .719 .265 QRE
b λ bP R ) Avg(X bP R ) Q1(X bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
18.9 .438 .350 .450 .500 .539 .338 -1810.6 -1758.3 -52.3
20.9 .522 .480 .550 .600 .576 .377 -1782.2 -1717.7 -64.5
21.7 .525 .490 .550 .600 .595 .356 -1841.2 -1763.2 -78.0
QGF b λ φb bP R ) Avg(X b Q1(XP R ) bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
20.6 1 .419 .350 .450 .500 .537 .315 -1770.0 -1717.7 -52.2
20.6 4 .516 .470 .540 .590 .573 .374 -1780.8 -1715.8 -65.0
20.3 1 .493 .450 .510 .560 .581 .336 -1818.0 -1736.8 -81.2
11.7 1 .591 .510 .600 .680 .419 .171 -2443.2 -2380.7 -62.4
Likelihood Ratio Test (P-Values) Overall Proposing Voting
0.0000 0.0000 0.7882
0.0905 0.0458 1.0000
0.0000 0.0000 1.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0532
0.0000 0.0000 1.0000
Note: Experiment 1 from Frechette, Kagel, and Lehrer (2003); Experiments 2 through 4 from Frechette, Kagel, and Morelli (2005b); Experiment 5 from Frechette, Kagel, and Morelli (2005a); Experiment 6 ? , and X bP R refer, respectively to the proposer’s from Drouvelis, Montero, and Sefton (2010). XP R , XP R allocation observed in the data, the proposer’s allocation predicted by the benchmark model, and the proposer’s allocation predicted by the MLE estimates. % MWC refers to the incidence of minimum winning coalitions, defined as proposals where at least 1 member (for n = 3), or at least 2 members (for n = 5), receive less than 5% of the pie. All data and estimates refer to round 1 behavior.
26
Table 3: Estimation results, round 1 data, experienced subject only Experiment
1
2
3
4
5
6
N δ Observations XP? R
5 4/5 75 .680
3 1 90 .666
3 1 162 .666
3 1/2 120 .833
5 1 150 .600
3 1 240 .666
.588 .530 .580 .670 .858 .050
.419 .350 .400 .450 .947 .500
.507 .460 .500 .580 .729 .350
15.3 .643 .580 .650 .710 .494 .134 -891.0 -878.5 -12.5
36.0 .414 .400 .450 .450 .823 .298 -907.7 -871.7 -36.0
21.2 .523 .480 .550 .600 .583 .369 -1812.4 -1722.0 -90.4
37.0 1 .400 .350 .400 .450 .855 .255 -873.9 -840.1 -33.8
21.8 1 .496 .450 .510 .560 .620 .292 -1747.6 -1654.0 -93.6
Data Avg(XP R ) Q1(XP R ) Q2(XP R ) Q3(XP R ) % MWC P[delay]
.366 .350 .400 .400 .653 .000
.556 .530 .570 .570 .800 .233
.525 .500 .530 .570 .907 .148 QRE
b λ bP R ) Avg(X bP R ) Q1(X bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
23.4 .447 .400 .450 .500 .587 .287 -617.9 -604.9 -13.0
25.6 .540 .510 .560 .600 .684 .263 -591.4 -576.0 -15.4
26.3 .543 .510 .560 .600 .698 .249 -1052.3 -1028.8 -23.5
QGF b λ φb bP R ) Avg(X b Q1(XP R ) bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
24.4 1 .427 .350 .450 .500 .604 .267 -599.8 -586.0 -13.8
25.1 3 .530 .500 .540 .580 .680 .257 -589.7 -574.2 -15.5
28.1 1 .505 .470 .510 .550 .753 .166 -983.8 -955.6 -28.2
18.4 1 .611 .560 .620 .670 .575 .106 -838.5 -828.8 -9.7
Likelihood Ratio Test (P-Values) Overall Proposing Voting
0.0000 0.0000 1.0000
0.0619 0.0580 1.0000
0.0000 0.0000 1.0000
0.0000 0.0000 0.0178
0.0000 0.0000 0.0359
0.0000 0.0000 1.0000
Note: Experiment 1 from Frechette, Kagel, and Lehrer (2003); Experiments 2 through 4 from Frechette, Kagel, and Morelli (2005b); Experiment 5 from Frechette, Kagel, and Morelli (2005a); Experiment 6 ? , and X bP R refer, respectively to the proposer’s from Drouvelis, Montero, and Sefton (2010). XP R , XP R allocation observed in the data, the proposer’s allocation predicted by the benchmark model, and the proposer’s allocation predicted by the MLE estimates. % MWC refers to the incidence of minimum winning coalitions, defined as proposals where at least 1 member (for n = 3), or at least 2 members (for n = 5), receive less than 5% of the pie. All data and estimates refer to round 1 behavior.
27
Table 4: Estimation results, all data Experiment
1
2
3
4
5
6
N δ Observations XP? R
5 4/5 285 .680
3 1 486 .666
3 1 555 .666
3 1/2 474 .833
5 1 745 .600
3 1 678 .666
.538 .500 .530 .600 .679 .093
.405 .350 .400 .500 .840 .422
.491 .420 .500 .580 .603 .269
9.7 .622 .530 .640 .730 .416 .189 -4063.1 -3897.8 -165.2
29.2 .376 .350 .400 .450 .680 .459 -5582.7 -5286.3 -296.4
16.1 .504 .430 .530 .610 .465 .499 -5715.2 -5372.6 -342.6
29.1 1 .369 .300 .400 .450 .682 .434 -5541.0 -5237.6 -303.3
16.1 1 .495 .420 .520 .600 .469 .464 -5663.5 -5299.0 -364.5
Data Avg(XP R ) Q1(XP R ) Q2(XP R ) Q3(XP R ) % MWC P[delay]
.333 .250 .350 .400 .407 .036
.556 .500 .530 .600 .739 .318
.520 .500 .500 .570 .791 .219 QRE
b λ bP R ) Avg(X bP R ) Q1(X bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
19.4 .439 .350 .450 .500 .541 .333 -2567.7 -2488.8 -78.9
18.1 .511 .460 .540 .600 .509 .451 -3828.9 -3626.7 -202.2
19.8 .517 .470 .540 .600 .550 .406 -4231.7 -4015.3 -216.4
QGF b λ φb bP R ) Avg(X b Q1(XP R ) bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
20.9 1 .427 .350 .450 .500 .545 .312 -2511.5 -2431.6 -79.9
18.0 6 .509 .450 .540 .600 .508 .448 -3828.4 -3625.2 -203.3
19.2 1 .500 .450 .520 .580 .546 .370 -4181.5 -3950.9 -230.5
11.9 1 .599 .520 .610 .690 .425 .168 -3957.0 -3801.6 -155.4
Likelihood Ratio Test (P-Values) Overall Proposing Voting
0.0000 0.0000 1.0000
0.3390 0.0808 1.0000
0.0000 0.0000 1.0000
0.0000 0.0000 0.0000
0.0000 0.0000 1.0000
0.0000 0.0000 1.0000
Note: Experiment 1 from Frechette, Kagel, and Lehrer (2003); Experiments 2 through 4 from Frechette, Kagel, and Morelli (2005b); Experiment 5 from Frechette, Kagel, and Morelli (2005a); Experiment 6 ? , and X bP R refer, respectively to the proposer’s from Drouvelis, Montero, and Sefton (2010). XP R , XP R allocation observed in the data, the proposer’s allocation predicted by the benchmark model, and the proposer’s allocation predicted by the MLE estimates. % MWC refers to the incidence of minimum winning coalitions, defined as proposals where at least 1 member (for n = 3), or at least 2 members (for n = 5), receive less than 5% of the pie. All data and estimates refer to all rounds of behavior.
28
Table 5: Estimation results, paths of play Experiment
1
2
3
4
5
6
N δ Observations XP? R
5 4/5 57/55 .680
3 1 162/110 .666
3 1 185/137 .666
3 1/2 158/140 .833
5 1 149/90 .600
3 1 226/160 .666
.416 .350 .400 .500 .819 .422
.495 .420 .500 .580 .655 .269
32.5 .395 .350 .400 .450 .751 .377 -1060.2
19.7 .517 .470 .540 .600 .547 .409 -1784.9
32.5 1 .386 .350 .400 .450 .760 .345 -1047.5
20.4 1 .502 .450 .520 .580 .577 .333 -1739.3
Data Avg(XP R ) Q1(XP R ) Q2(XP R ) Q3(XP R ) % MWC P[delay]
.323 .200 .350 .400 .421 .036
.556 .500 .530 .600 .716 .318
.525 .500 .500 .570 .805 .219
.552 .500 .530 .600 .709 .093 QRE
b λ bP R ) Avg(X bP R ) Q1(X bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall
20.0 .440 .350 .450 .500 .545 .326 -507.4
20.5 .520 .480 .540 .600 .567 .387 -1248.5
23.5 .533 .500 .550 .600 .638 .310 -1334.5
10.7 .627 .540 .640 .730 .425 .178 -1285.3 QGF
b λ φb bP R ) Avg(X b Q1(XP R ) bP R ) Q2(X bP R ) Q3(X % MWC P[delay] Ln(L) – Overall
21.7 1 .428 .350 .450 .500 .555 .301 -493.2
19.5 1 .501 .450 .520 .580 .554 .360 -1240.9
23.3 1 .507 .470 .520 .570 .650 .253 -1296.8
13.1 1 .600 .530 .610 .680 .446 .154 -1243.7
Likelihood Ratio Test (P-Values) Overall
0.0000
0.0001
0.0000
0.0000
0.0000
0.0000
Note: Experiment 1 from Frechette, Kagel, and Lehrer (2003); Experiments 2 through 4 from Frechette, Kagel, and Morelli (2005b); Experiment 5 from Frechette, Kagel, and Morelli (2005a); Experiment 6 ? , and X bP R refer, respectively to the proposer’s from Drouvelis, Montero, and Sefton (2010). XP R , XP R allocation observed in the data, the proposer’s allocation predicted by the benchmark model, and the proposer’s allocation predicted by the MLE estimates. % MWC refers to the incidence of minimum winning coalitions, defined as proposals where at least 1 member (for n = 3), or at least 2 members (for n = 5), receive less than 5% of the pie. All data and estimates refer to observed paths of play, i.e., selected proposals only. Number of observations overall/number of paths.
29
7
Log-likelihood of SSPE benchmark
The table below displays log-likelihood and summary statistics for two models. b estimates. The second model is The first model is QRE at the benchmark λ b QRE at λ = 500, a close approximation of the benchmark SSPE. The bottom panel of the table shows results of a likelihood ratio test of a null that both models fit the experimental data equally well. b vs. SSPE (QRE at λ b = 500) Table 6: Likelihood ratio test: QRE at benchmark λ Experiment
1
2
3
4
5
6
N δ Observations XP? R
5 4/5 275 .680
3 1 330 .666
3 1 411 .666
3 1/2 420 .833
5 1 450 .600
3 1 480 .666
.537 .695 .093
.409 .853 .422
.486 .573 .269
10.6 .627 .424 .179 -3432.5 -3346.2 -86.3
33.5 .400 .773 .353 -3012.7 -2920.9 -91.8
18.4 .512 .516 .443 -3849.9 -3662.4 -187.6
500 .500 1.00 .000 -32182.9 -31350.0 -832.9
500 .653 1.00 .012 -51178.1 -49805.8 -1372.3
Data Avg(XP R ) % MWC P[delay]
.338 .422 .036
.553 .727 .318
.522 .793 .219 QRE
b λ bP R ) Avg(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
20.2 .441 .547 .324 -2434.1 -2369.3 -64.8
22.1 .527 .605 .345 -2380.4 -2300.1 -80.2
23.4 .532 .635 .313 -2903.3 -2800.7 -102.7
b SSPE (large λ) b λ bP R ) Avg(X % MWC P[delay] Ln(L) – Overall Ln(L) – Proposing Ln(L) – Voting
500 .600 1.00 .000 -40715.2 -40443.9 -271.3
500 .653 1.00 .012 -30962.3 -30422.0 -540.3
500 .653 1.00 .012 -35872.4 -35139.8 -732.5
500 .820 1.00 .001 -65342.3 -64611.7 -730.6
Likelihood Ratio Test (P-Values) Overall Proposing Voting
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
Note: Experiment 1 from Frechette, Kagel, and Lehrer (2003); Experiments 2 through 4 from Frechette, Kagel, and Morelli (2005b); Experiment 5 from Frechette, Kagel, and Morelli (2005a); Experiment 6 ? , and X bP R refer, respectively to the proposer’s from Drouvelis, Montero, and Sefton (2010). XP R , XP R allocation observed in the data, the proposer’s allocation predicted by the benchmark model, and the proposer’s allocation predicted by the MLE estimates. % MWC refers to the incidence of minimum winning coalitions, defined as proposals where at least 1 member (for n = 3), or at least 2 members (for n = 5), receive less than 5% of the pie. All data and estimates refer to round 1 behavior.
30
8
Proposals Distribution and Acceptance Probabilities in QRE
The first three figures below show the pdf of proposals made by player 3 in the QRE with n = 3 and δ = 1 for λ ∈ {2, 10, 32}. The second three figures show the probability of acceptance for the same parameters. The location of the vertical lines in the figures corresponds to the SSPE in the benchmark model.
31
Figure 20: QRE proposal pdf N = 3, δ = 1, λ = 2
3 · 10−6 2 · 10−6 1 · 10−6
0 1
1 0.75
0.75 0.5 x2
0.5 0.25
0.25 0 0
32
x1
Figure 21: QRE proposal pdf N = 3, δ = 1, λ = 10
8 · 10−6 6 · 10−6 4 · 10−6 2 · 10−6 0 1
1 0.75
0.75 0.5 x2
0.5 0.25
0.25 0 0
33
x1
Figure 22: QRE proposal pdf N = 3, δ = 1, λ = 32
1.5 · 10−4
1 · 10−4
5 · 10−5
0 1
1 0.75
0.75 0.5 x2
0.5 0.25
0.25 0 0
34
x1
Figure 23: QRE probability of acceptance N = 3, δ = 1, λ = 2
0.4
0.2
0 1
1 0.75
0.75 0.5 x2
0.5 0.25
0.25 0 0
35
x1
Figure 24: QRE probability of acceptance N = 3, δ = 1, λ = 10
0.6
0.4
0.2
0 1
1 0.75
0.75 0.5 x2
0.5 0.25
0.25 0 0
36
x1
Figure 25: QRE probability of acceptance N = 3, δ = 1, λ = 32
1
0.5
0 1
1 0.75
0.75 0.5 x2
0.5 0.25
0.25 0 0
37
x1
9
Proposals Distribution in Experimental Data
The six figures below show the distribution of the experimental round 1 proposals. The data has been reorganized such that player 1 is the proposer, hence the figures show allocations to players 2 and 3. For experiments with n = 5 we summed x2 + x3 and x4 + x5 in order to be able to plot the data, where, for a given observation, x2 and x3 (x4 and x5 ) denote the two highest (lowest) allocations to the non-proposing players. The distribution has been rescaled to be symmetric around the x2 = x3 or x2 + x3 = x4 + x5 axis. The location of the vertical lines in the figures corresponds to the SSPE in the benchmark model.
38
Figure 26: Experiment 1, round 1 proposal distribution, proposing player 1
0.2
0.15
0.1 5 · 10−2 0 1
1 0.75
0.75 0.5
x4 + x5
0.5 0.25
0.25 0 0
39
x2 + x3
Figure 27: Experiment 2, round 1 proposal distribution, proposing player 1
0.1
5 · 10−2
0 1
1 0.75
0.75 0.5 x3
0.5 0.25
0.25 0 0
40
x2
Figure 28: Experiment 3, round 1 proposal distribution, proposing player 1
0.15
0.1
5 · 10−2
0 1
1 0.75
0.75 0.5 x3
0.5 0.25
0.25 0 0
41
x2
Figure 29: Experiment 4, round 1 proposal distribution, proposing player 1
8 · 10−2 6 · 10−2 4 · 10−2 2 · 10−2 0 1
1 0.75
0.75 0.5 x3
0.5 0.25
0.25 0 0
42
x2
Figure 30: Experiment 5, round 1 proposal distribution, proposing player 1
0.15
0.1
5 · 10−2
0 1
1 0.75
0.75 0.5
x4 + x5
0.5 0.25
0.25 0 0
43
x2 + x3
Figure 31: Experiment 6, round 1 proposal distribution, proposing player 1
0.15
0.1
5 · 10−2
0 1
1 0.75
0.75 0.5 x3
0.5 0.25
0.25 0 0
44
x2
10
Stationary Version of Gambler’s Fallacy Model
We can model Gambler’s Fallacy with an alternative assumption: with probability φ ∈ [0, 1] the current proposer is excluded from the proposer recognitions in the following period and with probability 1 − φ the current proposer is included in the proposer recognitions in the following period. The rest of the model is as described in the paper. The structure of the equilibrium clearly remains the same as in the benchmark model. The proposer allocates nothing to n−1 2 players, allocates a share players and keeps what remains for herself. that ensures a positive vote to n−1 2 Denote by x the equilibrium share a responder receives from the proposer if she is a member of the coalition of players supporting the proposal. Given the equilibrium continuation value of the respondents, vr , we have x = δvr . A player’s continuation value conditional on being recognized to propose, vp , is: 1 n−1 n−11 1 1− x + x vp = φ x + (1 − φ) 2 n 2 n 2 (1) 2 + φ(xn − 2) = 2n A player’s continuation value conditional on not being recognized to propose, vr , is 1 n−1 n−21 vr = φ 1− x + x n−1 2 n−12 n−1 n−11 1 (2) 1− x + x + (1 − φ) n 2 n 2 φ(1 − x2 ) 1 − φ = + . n−1 n Solving x = δvr for x gives x=
δ 2n − 2 + 2φ n 2n − 2 + δφ
which equals nδ when φ = 0 (as in the benchmark model) and is clearly increasing in φ. The last thing we need to check is that 1 − n−1 2 x > δvp , that is, the proposer attains a higher payoff by allocating x to n−1 players rather than proposing 2 an allocation which would be rejected and then receiving, in expectation, δvp . Substituting the expression for vp from above into 1 − n−1 2 x > δvp gives x< which combined with x =
2 n + δφ − δ n n + δφ − 1
2δ n−1+φ n 2n−2+δφ
rewrites as
n(n(2 − δ) − 2) + δφ(n(2 − δ) − 1) + δ > 0 which clearly holds since n(2 − δ) ≥ 3 for any δ ∈ [0, 1] and n ∈ N≥3 .
45
References Baron, D. P. and J. A. Ferejohn (1989). Bargaining in legislatures. American Political Science Review 83 (4), 1181–1206. Drouvelis, M., M. Montero, and M. Sefton (2010). Gaining power through enlargement: Strategic foundations and experimental evidence. Games and Economic Behavior 69 (2), 274–292. Frechette, G. R., J. H. Kagel, and S. F. Lehrer (2003). Bargaining in legislatures: An experimental investigation of open versus closed amendment rules. American Political Science Review 97 (2), 221–232. Frechette, G. R., J. H. Kagel, and M. Morelli (2005a). Behavioral identification in coalitional bargaining: An experimental analysis of demand bargaining and alternating offers. Econometrica 73 (6), 1893–1937. Frechette, G. R., J. H. Kagel, and M. Morelli (2005b). Nominal bargaining power, selection protocol, and discounting in legislative bargaining. Journal of Public Economics 89 (8), 1497–1517.
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