The role of network structure in the diffusion of investment decisions 3
Model
Furthermore, when the state is high, agents will prefer to make the investment sooner rather than later. Consequently, we will say that a family of bounded networks(𝐺𝑛) hinders diffusion if there is a ¯𝑝 < 1 such that the probability of any agent making the right investment decision in the long run in any equilibrium at each𝐺𝑛 for each value of the discount factor 𝛿 is at most ¯𝑝.
Results
That is, agent 𝑖 makes the right investment decision in the long run if she ultimately invests in the high state and never invests in the low state. In particular, this implies that there is no universal limit on the probability that agents will make the right investment decision in long-run equilibrium (ultimately investing in the high state, never investing in the low state).
Equilibria on the line
In this case, agent𝑖+1 learns relatively little by observing that agent𝑖 chooses not to invest in period𝑡. It is possible that in such a scenario, the agent𝑖 chooses not to invest in period 𝑡 because she is genuinely pessimistic about the prospects of investing.
Conclusion
Note that under any strategy profile, each agent's expected payoff is at most 12, since each agent can have a payoff of at most 0 in the low state and 1 in the high state. For any 𝜖 > 0, there is a 𝛿 < 1 and a symmetric strategy profile on the line under which each agent's expected payoff is at least 12 −𝜖.
Setting up the optimal stopping rule problem
Note that given the other agents' strategies, agent𝑖's problem is to find an optimal stopping rule𝜏𝑖. Therefore, agent𝑖's problem is to find an optimal stopping rule with respect to the flow of signals produced by her neighbors' actions, conditional on her not investing.
Optimal stopping rules are always threshold strategies
If the agent𝑖adopts at some point𝑡, it means that she observed how her neighbors behaved in periods 0 through 𝑡−1 in response to her not investing in those periods; in particular, it means that the actions her neighbors take after she invests do not play any role in her decisions. Since 𝑖 effectively leaves the game as soon as she invests, 𝑠𝑖's problem, given her private signal 𝑠𝑖, is to find an optimal stopping rule with respect to the signal flow (ℎ˜𝑖.
Families of unbounded degree
No spontaneous investment on trees
But by the lemma, this would mean that the behavior is suboptimal with non-zero. period𝑡−1, agent𝑖 does not invest in period𝑡, and we will say that a strategy profile satisfies no spontaneous investment if every agent's strategy satisfies no spontaneous investment. Another useful fact is that if 𝐺 is a tree, then seeing a neighbor invest is always evidence that the state is high.
Making the right investment decision at infinity as a spectator
Dynamics on the line
We are now ready to show that the decisions of agent𝑖+1 cannot be arbitrarily informative for agent𝑖. By fully symmetric arguments, it also says that the decisions of agent𝑖−1 cannot be arbitrarily likely to be arbitrarily informative for agent𝑖.
Outside observer learns from period 0 decisions
No physical impediments on the line
Proofs of lemmas
As it turns out, if 𝑗 is in the orbit of 𝑖, then there is a set 𝑔 ∈𝐺 such that 𝑔(𝑖) = 𝑗, the coset of the stabilizer 𝐺𝑖. Since the voting rule is fair, all functions 𝑓𝑖 are isomorphic and thus have minimal winning coalitions of the same size.
The speed of sequential asymptotic learning
Introduction
Here we calculate exactly the asymptotic behavior of the log-likelihood ratio of public trust. Chamley Chamley (2004b) provides an estimate of the evolution of public trust for a class of fat-tailed private signal distributions.
Model
The evolution of public belief
We next move on to more precisely estimate the long-run behavior of the public log-likelihood ratioℓ𝑡. 7By "the left tail of 𝐺−is convex and differentiable" we mean that there are some𝑥0 which, restricted to (−∞, 𝑥0),𝐺−is convex and differentiable. In particular,𝑇1 has infinite expectation, and thus, since𝑇𝐿 > 𝑇1, the expectation of the time to learn𝑇𝐿 is also infinite.
As we noted above, the time to learn can only be large if at least one of the runs is long.
Conclusion
Sub-linear learning
This is due to the fact that if an outside observer believing 𝜇𝑡 had to choose an action, he would choose 𝑎𝑡−1, the action of the last player he observed, the player who strictly has more information than she does. Then ℓ𝑡 is positive with probability 1 from some point onwards and all agents take action+1 from that point onwards. Given 𝑟𝑡, we will construct private signal distributions such that lim inf𝑡|ℓ𝑡|/𝑟𝑡 > 0 with probability one.
Intuitively, if we can choose private signal distributions that cause 𝐷+(𝑥) to decay very slowly, then ℓ𝑡 will be very close to being linear.
Long-term behavior of public belief
To show that (1−𝜀)𝑎𝑡 ≤ 𝑏𝑡 ≤ (1+𝜀)𝑎𝑡 holds for infinitely many values of 𝑡, let 𝑥0> 1 such that for all 𝑥 > 𝑥0, 𝐴and 𝐵 are monotonically decreasing. 2.7) Assume that 𝑎𝑡, 𝑏𝑡 > 𝑥0; this will indeed be considered large enough since 𝐴 and 𝐵 are positive and continuous, so both 𝑎𝑡 and 𝑏𝑡 are monotonically increasing and tending to infinity. Here we show that if (2.6) holds for sufficiently large𝑡, then it holds for all𝑡′> 𝑡. Note that 𝑎(𝑡) and 𝑏(𝑡) increase monotonically with time and tend to infinity as 𝑡 tends to infinity.
We end this section with a lemma showing that under certain technical conditions on the left side of 𝐺− the function 𝑢+(𝑥) = 𝑥 + 𝐷+(𝑥) (i.e. the function that determines how the public log-likelihood ratio is updated when action +1 is taken) eventually increases monotonically.
Gaussian private signals
Denote by 𝐸𝑡 the event that 𝑎𝜏 = +1 for all 𝜏 ≥ 𝑡; that is, over time there are no more errors𝑡. The following lemma gives a uniform bound for the probability of 𝐸𝑡 depending on public belief. Furthermore, it follows from a routine application of L'Hopital's rule (or from the standard asymptotic expansion for the CDF of a normal distribution) that is sufficiently large for everyone.
Calculate at 𝐶𝑡 being the event that 𝑎𝜏 =+1 for all 𝜏 < 𝑡, and note that the event𝑇1=𝑡 is simply the intersection of 𝐶𝑡 and the event that𝑎𝑡 =−1.
Upsets and runs
In particular, the probability that Ξ𝑡 is logarithmic in 𝑡 tends to zero as 𝑡 tends to infinity. An important consequence of Corollary 2 is that with high probability, there is at least one maximal run before time 𝑡 which is long relative to 𝑡. Then there exists a 𝑧 > 0 such that, if there is a good length 𝑠of𝑡, thenℓ𝑡+𝑠 ≥ ℓ∗.
Furthermore, since 𝑢+(𝑥) > 𝑥 for all𝑥, it follows that whenever there is a series of lengths 𝑁 from 𝑡, ℓ𝑡+𝑁 > ℓ∗.
Distributions with polynomial tails
That is, Aut𝑓 is the set of permutations of the voters that leave the election result unchanged, for each voting profile. In words, a voting rule is fair if, for two voters 𝑣 and 𝑤, there is some permutation of the population that relabels𝑣 as 𝑤 such that regardless of the voters'. The longest vote rule is fair since any rotation of the cycle is an automorphism.
Thus, if 𝑅 is a subgroup of the automorphism group of a voting rule 𝑓, then this rule is cyclic if 𝑛 = 𝑝, and is either cyclic, 2-cyclic, or both if 𝑛 = 𝑝2.
Equitable voting rules
Introduction
We analyze the winning coalitions of such fair voting rules and show that they can comprise a vanishing fraction of the population, but not less than the square root of its size. Each district elects its representative by majority rule, and elections are decided by majority rule of representatives. In a representative democracy, winning coalitions must comprise at least a quarter of the population.3 Much smaller winning coalitions are possible in voting rules called generalized representative democracy (GRD), where voters are hierarchically divided into groups, which are then divided into subgroups and so on.
For each group, the outcome is given by majority rule over subgroup decisions.4 We show that equal GRD rules for voters can have winning coalitions as small as When the number of voters𝑛is a perfect square, and when commission sizes are taken as√.
The model
Given an abstract voting rule 𝑓: 🝑 −1). A different assignment with An abstract voting rule 𝑓: 𝑋𝑅 → 𝑋 is symmetric if and only if all assignments are equivalent under 𝑓.
An abstract voting rule 𝑓: 𝑋𝑅 → 𝑋 is fair if and only if all roles are equivalent under 𝑓.
Winning coalitions for equitable voting rules
Conversely, for arbitrarily large votes, fair general voting rules exist for representative democracy with winning coalitions of size log32. In order to form a winning coalition, two of the three top representatives must agree with each other. Then two of these representatives' voters must agree, and so on, recursively.
At each intermediate node, the results of the three nodes below are merged by majority.
There is an intriguing connection between this characteristic and the so-called Hausdorff dimension of the Cantor set, which is log. The connection arises from the fact that GRD rules with the smallest winning coalitions are those where the subdivision at each level is into three groups. As can be easily verified, none of the voting rules mentioned so far are examples, except for majority, 2-equal. We say that almost all natural numbers satisfy a property𝑃if the subset𝑁𝑃 ⊆N of the natural numbers that have property𝑃satisfies.
These results are a consequence of the successful completion of a large project involving thousands of articles and hundreds of authors, called the Classification of Finite Simple Groups, see Aschbacher et al.
Towards a characterization of equitable voting rules
Intuitively, a voting rule is 2-cyclic if one can arrange the voters on a 𝑛1× 𝑛2 grid such that all voters are shifted to the right (and the rightmost ones turn back to the left) or all voters are moved up (again the top fold .ones down) does not affect the outcome. It is easy to see that the cross-committee consensus rule is an example of such a rule, as is the representative democracy rule. This proposition has an important implication for the design of simple, fair voting rules that are not tailored to specific electorate sizes.
The majority rule is one such rule - one only needs to count the votes and consider the difference between the number of supporters of one alternative compared to the other.
Conclusions
In fact, such a rule must also work for electorates consisting of a prime number of electors, and in particular must be cyclic. It will be interesting to understand whether a much larger class of rules than cyclic voting rules can work for nearly all electorate sizes. In contrast, the set of rules for equitable suffrage is much richer, and its complexity is intimately linked to boundary problems in mathematics; especially the classification of finite simple groups.
This paper includes a number of different examples, including generalized representative democracy, intercommittee consensus, and a number of additional constructions that appear in the appendix, but the class of all equal rules is still larger.
A primer on finite groups
Interestingly, in his original treatise on games, Nash took an approach to symmetry similar to ours, by studying the automorphism group of the game. In our case, Aut𝑓 is the set of permutations of the voters that preserve each outcome of 𝑓. This is equivalent to there being only a single 𝐺 orbit, or 𝑗 being in the orbit of 𝑖 for every 𝑖, 𝑗.
Therefore, if 𝐺 is transitive, the orbit𝐺 𝑖 is of size 𝑛, and since we can identify this orbit with the adjuncts of𝐺𝑖, there are such adjuncts.
Proofs
A simple calculation shows that the winning coalition that recursively consists of the winning coalitions of two of {𝑉1, 𝑉2, 𝑉3} (e.g. 𝑉1 and𝑉3, as in Figure 3.4) is of size . Denote by 𝜂(𝑛) the number of positive integers 𝑚 ≤ 𝑛 for which there is no 2-transitive group action on a set of 𝑚elements other than 𝑆𝑚 and 𝐴𝑚. Taking 𝑅 to be the union of the cosets in 𝑃/𝑍 that include 𝑍′, 𝑅 acts transitively on𝑉.
That is, if 𝐺 is an abelian group of permutations of a set𝑉—and assume without loss of generality that no element of𝑉 is fixed by all elements of 𝐺—then there is a way to identify 𝑉 with Î𝑚.
