Single Peaked Preferences and the Median Voter Theorem

In the last entry on voting paradoxes, I mentioned that things are different if preferences are single peaked.

Let’s look at another scenario. Again, consider a set of policies A,B,C following a sequence from  (A) less to more extreme (C) - maybe tax rates or some level of spending- with (B) being the intermediate policy. Suppose voters rank policies in order of preference/utility as follows:

Voter X:  ABC
Voter Y : CBA
Voter Z : BCA

In this case, no matter what order is undertaken, B always ends up being the law that is enacted. These preferences are single peaked. Each individual has a most preferred choice along the A to C spectrum. If you move away from that choice (in the A-B-C spectrum) they prefer the other choices less.  In the previous example, voter Y did not have single peaked preferences and that is what caused the cycling or order dependent outcomes. With single peaked preferences there is a new problem. With single peaked preferences, the median point of the preference distribution will elicit the most votes. Only those laws or candidates with a centrist twist will get the majority of the votes. Only those voters with centrist views will be happy, and it makes it very difficult for candidates to be elected if they want to bring about major reforms. This phenomenon is referred to as the ‘median voter theorem.’

Both voter cycling (when preferences are not single peaked) and the median voter theorem can have negative implications for majority rule policy adoption. Voters are either governed by irrational ever changing majorities, or they are subjugated by entrenched majorities whose views are maintained by the status quo of median preferences.

References:

Lemieux, Pierre, The Public Choice Revolution. Regulation, Vol. 27, No. 3, pp. 22-29, Fall 2004. Available at SSRN: https://ssrn.com/abstract=604046

Majority Rule and Vote Cycling with Non-Single Peaked Preferences

Is majority rule the best way to represent voters preferences for a given set of policies?

Let’s look at a particular voting scenario to illustrate this. Consider a set of policies A,B,C following a sequence from  (A) less to more extreme (C) - maybe tax rates or some level of spending- with (B) being the intermediate policy. Suppose voters have the following preferences:


VOTER X: A >B >C  'single peaked '

VOTER Y: C>A>B

VOTER Z: B>C >A  'single peaked'

Both voters X and Z have single peaked preferences. As we move away from their optimal choice the strength of their preferences or utility decreases. However, voter Y does not exhibit single peaked preferences. They prefer the extreme policy C most, but their next preferred policy is in the direction of the other extreme A. They prefer the intermediate policy B least.

If the voters were voting on these issues, voter X would prefer law A over law B and law B over law C. In shorthand – A >B >C. To summarize all of the choices of the voters we see that 2/3 of the voters have preference A >B, 2/3 of the voters have preference B > C, but when voting A vs. C, 2/3 have preference C >A.

See if you follow the application of this. If we vote on policies in a pairwise fashion and have two elections and the first is made between policy B and C, then B will win (2/3 of the voters have preference B > C). If this is followed by a second election A vs. B (Because C was eliminated in the first election) then A will be the law that ultimately passes by majority rule.


Now if the order is changed, in which the first election is between A and B, A will win (because 2/3 of the voters rank A > B). Then in the second election when A goes against C, C will be the law that passes by majority rule (again because 2/3 of the voters have preference C >A).

So when voting on these policies, the process becomes arbitrary. The outcome depends on the order of the vote, so a cycling of choices ensues. According to public choice economist Gordon Tullock, any outcome can be obtained in majority voting by at least one voting method. This indicates that majorities can be irrational and dangerous unless preferences are all single peaked. 

Reference:

Lemieux, Pierre, The Public Choice Revolution. Regulation, Vol. 27, No. 3, pp. 22-29, Fall 2004. Available at SSRN: https://ssrn.com/abstract=604046

Risk, Uncertainty, Speculation and Granger Causality

What is the difference between 'risk' and 'uncertainty'?

"Uncertainty refers to outcomes that we cannot foresee, or whose probabilities that we cannot estimate. In other words, uncertainty is a way of characterizing what we don't know about the distribution of the random variables themselves...Risk can be quantified, priced,and traded. It can even be hedged with large pools of assets." - Froeb et al 2014.Managerial Economics: A problem solving approach. 3rd edition.

This distinction was first made by economist Frank Knight.

Insurance of all kinds and commodity futures markets are examples of financial products based on quantifiable risks. Those involved in these markets, often speculators, have an important role to play. Speculators and futures and options markets make it possible to allocate resources across time, essentially from periods of abundance to periods of scarcity smoothing consumption and alleviating risks.

This is described well in The Economic Way of Thinking:

"Speculators coordinate market exchanges through time...that both inform people and provide them with the opportunity to allocate their risks...Speculators accept the risk at a mutually agreed upon price that hedgers seek to avoid."

Elaine Kub, author of Mastering the Grain Markets puts it beautifully in her book:

"All life on earth depends on agriculture, how well we distribute agriculture's products-how well we trade grain-determines how Earth's population gains access to its most fundamental needs."

However, speculators and funds involved in commodities trading often come under heavy scrutiny perhaps without understanding the important role they play in actually increasing food security as Nevil Speer explains below:

“Reining in speculators seems politically expedient.  But we live in complex times.  Throwing darts becomes perilous when policy makers begin to advocate (and worse yet, actually believe) that speculators should be removed  from ag / food markets.   Such a move would dismantle futures markets.  Imagine what the world might look like a without market liquidity, price discovery and risk mitigation; not to mention the inability to establish pricing plans, attract new capital investment and stimulate innovation across the food business.   The absence of those influences, facilitated by futures markets, would ultimately lead to less food production, availability and security – NOT the other way around.  Taking speculators out of the mix would be devastating.”  Dr. Nevil Speer, No Speculators? No thanks!, Drovers Cattle Network Agsight, March 2011

So what impact have speculators had on markets? One way is to look at the impact of index funds on commodity prices and volatility. Economist Scott Irwin at the University of Illinois has looked at this in depth. Below are three examples of studies (although not a complete review of the literature) looking at these impacts based on granger causality tests and other methods:

The Impact of Index Funds in
Commodity Futures Markets:
A Systems Approach
DWIGHT R. SANDERS AND SCOTT H. IRWIN
The Journal of Alternative Investments
Summer 2011, Vol. 14, No. 1: pp. 40-49

"The system of Granger-style causality tests fails to reject the null hypothesis that that trader positions do not lead market returns. Hence, there is no evidence of a linkage between index trader positions in commodity futures markets and price levels."

Irwin, S. H. and D. R. Sanders (2010), “The Impact of Index and Swap Funds on Commodity Futures Markets: Preliminary Results”, OECD Food, Agriculture and Fisheries Working Papers, No. 27, OECD Publishing. doi: 10.1787/5kmd40wl1t5f-en

“There is no statistically significant relationship indicating that changes in index and swap fund positions have increased market volatility......at this time, the weight of evidence clearly suggests that increased index fund activity in 2006-08 did not cause a bubble in commodity futures prices.”

Index Trading and Agricultural Commodity Prices:
A Panel Granger Causality Analysis
Gunther Capelle-Blancard and Dramane Coulibaly
CEPII, WP No 2011 – 28
No 2011 – 28
December

"Our results show that, in agricultural futures markets, there is no evidence of a causality relationship from index funds to futures prices. This result holds for the period 2006-2010, but also for the sub-periods 2006-2008 and 2008-2010. These findings imply that index-based trading has not been an important driver in the substantial increase in commodities prices. Changes in commodity prices may instead reflect fundamental supply and demand factors."


References:

Explained: Knightian uncertainty
The economic crisis has revived an old philosophical idea about risk and uncertainty. But what is it, exactly?
Peter Dizikes, MIT News Office
June 2, 2010

Foeb, Mcann, Ward, and Shor. Managerial Economics: A problem solving approach. 3rd edition. 2008.

The Economic Way of Thinking. Heyne, Boettke, and Prychitko. 10th Edition. 2002.

Mastering the Grain Markets: How profits are really made. Elaine Kub.

See also: 

Fat Tails, Kurtosis, and Risk

Fat Tails, the Precautionary Principle and GMOs