Strategic Voting in Heterogeneous Electorates: An Experimental Study
Abstract
:1. Introduction
2. The Model
2.1. Theoretical Analysis
2. Behavioral Predictions
- (1)
- (2)
- Without information, there is no effect of heterogeneity on the probability of strategic voting (Figure 1 in comparison to TS11).
- (3)
- With information, there is no effect of heterogeneity on average behavior (Figure 3 in comparison to TS11).
- (4)
- Information about other’s intensities of preferences does not affect strategic behavior if preference orderings are known (Figure 3).
- (5)
- (6)
- With information, Rank 3rd voters vote more strategically than other Rank-Types Figure 4).
- (7)
- Heterogeneity decreases the probability of strategic voting of Rank 3rd voters with high intermediate value (Figure 4).
- (8)
- Heterogeneity increases the probability of strategic voting of Rank 2nd voters with high intermediate value (Figure 4).
3. Experimental Design
Preference Ordering | Intermediate Value = 3 | Intermediate Value = 8 | Total |
---|---|---|---|
A B C | 2 | 3 | 5 |
B C A | 1 | 2 | 3 |
C A B | 2 | 2 | 4 |
4. Results
4.1. Election Winner
4.2. Aggregate Behavior
4.3. Strategic Voting by Rank-Type
5. Concluding Remarks
Acknowledgments
Conflicts of Interest
References
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Appendix A. A Selection of Nash Equilibria
Uninformed
Aggregate Information
Full Information
CompositionGroup 1,2,3 | Group 1, um = 3 | Group 1, um = 8 | Group 2, um = 3 | Group 2, um = 8 | Group 3, um = 3 | Group 3, um = 8 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
SinCere | Strategic | Sincere | Strategic | Sincere | Strategic | Sincere | Strategic | Sincere | Strategic | Sincere | Strategic | |||
4 | 4 | 4 | 1.000 | - | 0.753 | 0.247 | 1.000 | - | 0.753 | 0.247 | 1.000 | - | 0.753 | 0.247 |
5 | 3 | 4 | 1.000 | 0.000 | 0.610 | 0.390 | 0.008 | 0.992 | - | 1.000 | 1.000 | - | 1.000 | - |
5 | 4 | 3 | 1.000 | 0.000 | 0.611 | 0.389 | 0.005 | 0.995 | - | 1.000 | 1.000 | - | 1.000 | - |
5 | 5 | 2 | 1.000 | 0.000 | 0.611 | 0.389 | 0.004 | 0.996 | - | 1.000 | 1.000 | - | 1.000 | - |
6 | 1 | 5 | 1.000 | - | 1.000 | - | - | 1.000 | - | 1.000 | 1.000 | - | 1.000 | - |
6 | 2 | 4 | 1.000 | - | 1.000 | - | - | 1.000 | - | 1.000 | 1.000 | - | 1.000 | - |
6 | 3 | 3 | 1.000 | - | 1.000 | - | - | 1.000 | - | 1.000 | 1.000 | - | 1.000 | - |
6 | 4 | 2 | 1.000 | - | 1.000 | - | - | 1.000 | - | 1.000 | 1.000 | - | 1.000 | - |
6 | 5 | 1 | 1.000 | - | 1.000 | - | - | 1.000 | - | 1.000 | 1.000 | - | 1.000 | - |
6 | 6 | 0 | 1.000 | - | 1.000 | - | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 |
7 | 0 | 5 | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
7 | 1 | 4 | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
7 | 2 | 3 | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
7 | 3 | 2 | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
7 | 4 | 1 | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
7 | 5 | 0 | 1.000 | - | 1.000 | - | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
8 | 0 | 4 | 1.000 | 0.000 | 1.000 | 0.000 | 0.333 | 0.333 | 0.333 | 0.333 | 0.335 | 0.334 | 0.338 | 0.337 |
8 | 1 | 3 | 1.000 | 0.000 | 1.000 | 0.000 | 0.343 | 0.329 | 0.343 | 0.329 | 0.335 | 0.334 | 0.337 | 0.337 |
8 | 2 | 2 | 1.000 | 0.000 | 1.000 | 0.000 | 0.343 | 0.329 | 0.343 | 0.329 | 0.335 | 0.334 | 0.337 | 0.337 |
8 | 3 | 1 | 1.000 | 0.000 | 1.000 | 0.000 | 0.343 | 0.329 | 0.342 | 0.329 | 0.335 | 0.334 | 0.337 | 0.337 |
8 | 4 | 0 | 1.000 | 0.000 | 1.000 | 0.000 | 0.342 | 0.329 | 0.342 | 0.329 | 0.333 | 0.333 | 0.333 | 0.333 |
9 | 0 | 3 | 0.999 | 0.000 | 0.993 | 0.006 | 0.333 | 0.333 | 0.333 | 0.333 | 0.359 | 0.346 | 0.389 | 0.390 |
9 | 1 | 2 | 0.999 | 0.000 | 0.994 | 0.005 | 0.421 | 0.293 | 0.416 | 0.301 | 0.357 | 0.342 | 0.381 | 0.379 |
9 | 2 | 1 | 0.999 | 0.000 | 0.995 | 0.004 | 0.411 | 0.298 | 0.406 | 0.306 | 0.356 | 0.340 | 0.375 | 0.372 |
9 | 3 | 0 | 0.999 | 0.000 | 0.995 | 0.003 | 0.403 | 0.302 | 0.398 | 0.311 | 0.333 | 0.333 | 0.333 | 0.333 |
10 | 0 | 2 | 0.997 | 0.000 | 0.975 | 0.020 | 0.333 | 0.333 | 0.333 | 0.333 | 0.393 | 0.367 | 0.436 | 0.461 |
10 | 1 | 1 | 0.997 | 0.000 | 0.979 | 0.015 | 0.483 | 0.268 | 0.471 | 0.288 | 0.389 | 0.355 | 0.429 | 0.435 |
10 | 2 | 0 | 0.996 | 0.000 | 0.982 | 0.012 | 0.466 | 0.276 | 0.453 | 0.298 | 0.333 | 0.333 | 0.333 | 0.333 |
11 | 0 | 1 | 0.994 | 0.000 | 0.954 | 0.034 | 0.333 | 0.333 | 0.333 | 0.333 | 0.424 | 0.382 | 0.452 | 0.503 |
11 | 1 | 0 | 0.992 | 0.000 | 0.961 | 0.026 | 0.515 | 0.257 | 0.495 | 0.289 | 0.333 | 0.333 | 0.333 | 0.333 |
12 | 0 | 0 | 0.988 | 0.000 | 0.935 | 0.044 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 | 0.333 |
Appendix B. Experimental Instructions
Welcome
Rounds and Decisions
Your Preference Ordering
In addition, at the start of every round, you will be informed how many participants in your electorate have been attributed to each of the three preference orderings. For example, you may hear that 5 voters have preference ordering A B C, 3 voters have B C A and 4 voters have C A B. You will not know others’ value for the middle option, however .[In addition, at the start of every round, you will be informed how many participants in your electorate have been attributed to each of the three preference orderings and how many points they will get for the middle option (X). For example, you may hear that 2 voters have preference ordering A B C with X=3 and 3 with X=8; 1 voter have B C A with X=3 and 2 with X=8 and 2 voters have C A B with X=3 and 2 with X=8.]
Trial Round
Appendix C. Random Draw Realizations
Election | ABC | BCA | CAB |
---|---|---|---|
1 | 4 | 5 | 3 |
2 | 1 | 4 | 7 |
3 | 3 | 5 | 4 |
4 | 3 | 4 | 5 |
5 | 2 | 6 | 4 |
6 | 7 | 2 | 3 |
7 | 6 | 3 | 3 |
8 | 4 | 5 | 3 |
9 | 3 | 6 | 3 |
10 | 1 | 7 | 4 |
11 | 5 | 1 | 6 |
12 | 6 | 4 | 2 |
13 | 4 | 3 | 5 |
14 | 3 | 3 | 6 |
15 | 2 | 9 | 1 |
16 | 4 | 2 | 6 |
17 | 7 | 3 | 2 |
18 | 2 | 4 | 6 |
19 | 4 | 1 | 7 |
20 | 3 | 1 | 8 |
21 | 4 | 4 | 4 |
22 | 5 | 5 | 2 |
23 | 2 | 5 | 5 |
24 | 4 | 4 | 4 |
25 | 3 | 4 | 5 |
26 | 4 | 5 | 3 |
27 | 4 | 3 | 5 |
28 | 2 | 6 | 4 |
29 | 5 | 4 | 3 |
30 | 4 | 3 | 5 |
31 | 2 | 4 | 6 |
32 | 8 | 1 | 3 |
33 | 2 | 7 | 3 |
34 | 2 | 6 | 4 |
35 | 3 | 5 | 4 |
36 | 5 | 5 | 2 |
37 | 10 | 2 | 0 |
38 | 5 | 1 | 6 |
39 | 2 | 2 | 8 |
40 | 5 | 5 | 2 |
Appendix D. Winning Probabilities
- 1Note, however, that preferences in TS11 are also heterogeneous in the sense that different voters have distinct induced preferences over outcomes. Nevertheless, we refer to preferences in TS11 as “homogenous” because voters attribute the same relative importance to the intermediate option.
- 2This is formalized as behavioral prediction #7, below.
- 3As an anonymous reviewer pointed out, the term ‘uninformed’ is somewhat inaccurate, because voters do have uniform priors over the preferences of others. We maintain the term to clearly label the three treatments.
- 5The principal branch of the Multinomial Logit Correspondence is defined and explained in [15]. In particular, see their Theorem 3. See our companion paper (TS11) for details on how the concept is applied to the problem at hand. Here, we note some of its advantages. First, it provides (in the limit) a refinement of Nash by providing a (generally) unique selection from the set of Nash equilibria (cf. Appendix A). Second, this selection is intuitive as it is based on the limit as behavioral noise reduces to zero. Third, the principal branch here has the intuitive characteristic that players of the same type play symmetric strategies.
- 6As pointed out by an anonymous reviewer, it may seem surprising that strategic voting can occur in equilibrium when voters have diffuse priors over others’ preferences. Note however that the Nash equilibrium involves no strategic voting. It is the introduction of noisy behavior (and responses to others’ noisy choices) that yields strategic voting. Moreover, a rational voter takes expectations over the possible realizations. For a voter with low intermediate value it is in most pivotal cases to her advantage to vote sincerely. For a voter with high intermediate value there are more cases in which strategic vote would be beneficial. This explains why the high value correspondence moves away more sharply from sincere voting.
- 7We deal with ties as follows. In case all three preference orderings are equally likely, all voters are ranked 1st. If two candidates have the same level of sincere support, the candidate (of these two) that is preferred by the supporters of the third candidate is ranked 1st. Supporters of the other candidate are ranked 2nd and the remaining voters are ranked 3rd. In case of a tie for second place, all voters for these two candidates are ranked 2nd.
- 8We plot the probabilities for μ ∈ [0,1]. For μ > 1, there is little difference across information treatments and all cases converge monotonically to 1/3.
- 10This is the (important) difference with TS11, where the payoff to the intermediate option was fixed in a session (either 3 or 8) and equal for all participants.
- 11Subjects that participated in the homogeneous sessions were not allowed to take part in the new set of sessions.
- 12There were 5 uninformed electorates, 6 electorates with aggregate information and 6 electorates with full information.
- 13Results for homogeneous settings (taken from TS11) are included for comparison. For these sessions, we average over treatments with low and high intermediate values.
- 14This result may seem somewhat counterintuitive, especially considering that there is more sincere voting in the uninformed case. Apparently, strategic votes increase the probability of the Majoritarian Candidate winning when voters are informed. In this sense information works as a coordination device (TS11). Given our interest in the effects of heterogeneity, further discussion of this result is beyond the scope of this paper, however.
- 15We obtain the following p-values for W, N = 12 compared to voters in the Full Information, Homogeneous Electorates. Aggregate Electorates,
= 3: p = 0.016; Aggregate Electorates,
= 8: p = 0.199; Full Information Electorates,
= 3: p = 0. 037; Full Information Electorates,
= 8: p = 0. 077.
- 16We obtain the following p-values for W, N = 12 compared to voters in the Full Information, Homogeneous Electorates. Aggregate Electorates,
= 3: p = 0.146; Aggregate Electorates,
= 8: p = 0.015, in the opposite direction; Full Information Electorates,
= 3: p = 0.053, Full Information Electorates,
= 8: p = 0.421.
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Tyszler, M.; Schram, A. Strategic Voting in Heterogeneous Electorates: An Experimental Study. Games 2013, 4, 624-647. https://doi.org/10.3390/g4040624
Tyszler M, Schram A. Strategic Voting in Heterogeneous Electorates: An Experimental Study. Games. 2013; 4(4):624-647. https://doi.org/10.3390/g4040624
Chicago/Turabian StyleTyszler, Marcelo, and Arthur Schram. 2013. "Strategic Voting in Heterogeneous Electorates: An Experimental Study" Games 4, no. 4: 624-647. https://doi.org/10.3390/g4040624
APA StyleTyszler, M., & Schram, A. (2013). Strategic Voting in Heterogeneous Electorates: An Experimental Study. Games, 4(4), 624-647. https://doi.org/10.3390/g4040624