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Figure from:Essays on Strategic Voting

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Figure 5

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Figure 2.1 Period-1 Equilibrium. We indicate the type of the juror 1 on the horizontal axis while the type of the juror 2 is measured on the vertical axis. We can represent any signal realization as a point in the coordinates. The cutoff types are shown on the axis. The realized verdict and the time of the decision are marked as well. This figure shows an equilibrium in which only juror 1’s information matters for the final verdict. She follows a strategy such that type xo is the critical type at the initial round. All the types that are lower than x9 say acquit no matter what the history is while all higher than xg say convict no matter what the history is. Any type of juror 2 gives in immediately. We think about this situation as a very poor way of aggregating the private information available for the jurors. Juror 1 states her opinion based on her own signal and is not willing to ‘listen’ to juror 2. As a best response, juror 2 does not reveal information but terminates the game immediately. Therefore, juror 1’s strategy is best response. (y; and y_, are the critical types for the second juror if she is a C- or an A-juror, respectively). Notice that here yj = “highest possible type” exactly means that all the possible types of juror 2 give in for acquit immediately if juror 1 voted for acquit initially. We can interpret y_; = lowest possible type similarly, i.e. all the possible types of juror 2 gives in for convict if juror 1 voted so.
Figure 2.1 Period-1 Equilibrium. We indicate the type of the juror 1 on the horizontal axis while the type of the juror 2 is measured on the vertical axis. We can represent any signal realization as a point in the coordinates. The cutoff types are shown on the axis. The realized verdict and the time of the decision are marked as well. This figure shows an equilibrium in which only juror 1’s information matters for the final verdict. She follows a strategy such that type xo is the critical type at the initial round. All the types that are lower than x9 say acquit no matter what the history is while all higher than xg say convict no matter what the history is. Any type of juror 2 gives in immediately. We think about this situation as a very poor way of aggregating the private information available for the jurors. Juror 1 states her opinion based on her own signal and is not willing to ‘listen’ to juror 2. As a best response, juror 2 does not reveal information but terminates the game immediately. Therefore, juror 1’s strategy is best response. (y; and y_, are the critical types for the second juror if she is a C- or an A-juror, respectively). Notice that here yj = “highest possible type” exactly means that all the possible types of juror 2 give in for acquit immediately if juror 1 voted for acquit initially. We can interpret y_; = lowest possible type similarly, i.e. all the possible types of juror 2 gives in for convict if juror 1 voted so.
Figure 2.3 Period-3 Equilibrium. This figure shows yet another improvement of the previous equilibrium. Here, the juror 1’s strategy reveals more information than before. Type xo is the critical type at period 0 and all the types in the range [x2,x9] can be convinced by the action of the juror 2. Furthermore, all the types who are more extreme then x2 insists on A. Concerning the juror 2, some types (ones with higher posterior than y;) hold out in period 1 and give in at period 3 if the other juror has not yet done so.
Figure 2.3 Period-3 Equilibrium. This figure shows yet another improvement of the previous equilibrium. Here, the juror 1’s strategy reveals more information than before. Type xo is the critical type at period 0 and all the types in the range [x2,x9] can be convinced by the action of the juror 2. Furthermore, all the types who are more extreme then x2 insists on A. Concerning the juror 2, some types (ones with higher posterior than y;) hold out in period 1 and give in at period 3 if the other juror has not yet done so.
The efficient decision rule in this joint decision problem is:
The efficient decision rule in this joint decision problem is:
on the horizontal and the density on the vertical axes). more extreme voters (with z close to —1 or 1) under distribution F than under G. The nex can generally be the case that there are both more indifferent voters (with z close to 0) and that this defines solely a partial order on distributions. For arbitrary distributions F, G it
on the horizontal and the density on the vertical axes). more extreme voters (with z close to —1 or 1) under distribution F than under G. The nex can generally be the case that there are both more indifferent voters (with z close to 0) and that this defines solely a partial order on distributions. For arbitrary distributions F, G it
and substitute these expressions back into the probabilities of being pivotal: The Effect of the Average Margin on the Pivotal Probabilities. With respect to the
and substitute these expressions back into the probabilities of being pivotal: The Effect of the Average Margin on the Pivotal Probabilities. With respect to the
The second graph suggests that the pivotal probability becomes symmetric to the margin as the size of the electorate increases. I plot the pivotal probability of a right voter as a function of the expected margin for electorate sizes N = 50 and N = 100. The curves indicating higher pivotal probabilities gives the values for the population with size N = 50 To indicate asymmetry, I included the pivotal probability of a left voter here, marked with dashed and dotted lines.
The second graph suggests that the pivotal probability becomes symmetric to the margin as the size of the electorate increases. I plot the pivotal probability of a right voter as a function of the expected margin for electorate sizes N = 50 and N = 100. The curves indicating higher pivotal probabilities gives the values for the population with size N = 50 To indicate asymmetry, I included the pivotal probability of a left voter here, marked with dashed and dotted lines.
The Effect of the Average Turnout on the Pivotal Probabilities. I assume that the . dashed and dotted lines. To indicate asymmetry, I included the pivotal probability of a left voter here, marked with
The Effect of the Average Turnout on the Pivotal Probabilities. I assume that the . dashed and dotted lines. To indicate asymmetry, I included the pivotal probability of a left voter here, marked with
Numerical computations show that (i) the positive effect disappears for zero margin put (ii) for high absolute value of margin, the positive effect overtakes as the size of the
Numerical computations show that (i) the positive effect disappears for zero margin put (ii) for high absolute value of margin, the positive effect overtakes as the size of the