single-crossing condition

Cumulative distribution functions F and G satisfy the single-crossing condition if there exists a $y^*$ such that

$\backslash forall\; x,\; x\; \backslash ge\; y^*\; \backslash implies\; F(x)\; \backslash ge\; G(x)$

and

$\backslash forall\; x,\; x\; \backslash le\; y^*\; \backslash implies\; F(x)\; \backslash le\; G(x)$;

that is, function $h(x)\; =\; F(x)-G(x)$ crosses the x-axis at most once, in which case it does so from below.

This property can be extended to two or more variables.{{Cite journal |last=Rösler |first=Uwe |date=September 1992 |title=A fixed point theorem for distributions |journal=Stochastic Processes and Their Applications |language=en |volume=42 |issue=2 |pages=195–214 |doi=10.1016/0304-4149(92)90035-O|doi-access=free }} Given x and t, for all x'>x, t'>t,

$F(x\text{'},t)\; \backslash ge\; F(x,t)\; \backslash implies\; F(x\text{'},t\text{'})\; \backslash ge\; F(x,t\text{'})$

and

$F(x\text{'},t)\; >\; F(x,t)\; \backslash implies\; F(x\text{'},t\text{'})\; >\; F(x,t\text{'})$.

This condition could be interpreted as saying that for x'>x, the function g(t)=F(x',t)-F(x,t) crosses the horizontal axis at most once, and from below. The condition is not symmetric in the variables (i.e., we cannot switch x and t in the definition; the necessary inequality in the first argument is weak, while the inequality in the second argument is strict).

In social choice theory, the single-crossing condition is a condition on preferences. It is especially useful because utility functions are generally increasing (i.e. the assumption that an agent will prefer or at least consider equivalent two dollars to one dollar is unobjectionable).{{Cite journal |last=Jewitt |first=Ian |date=January 1987 |title=Risk Aversion and the Choice Between Risky Prospects: The Preservation of Comparative Statics Results |url=https://academic.oup.com/restud/article-lookup/doi/10.2307/2297447 |journal=The Review of Economic Studies |volume=54 |issue=1 |pages=73–85 |doi=10.2307/2297447|jstor=2297447 }}

Specifically, a set of agents with some unidimensional characteristic $\backslash alpha^i$ and preferences over different policies q satisfy the single crossing property when the following is true:

If $q\; >\; q\text{'}$ and $\backslash alpha^\{i\text{'}\}\; >\; \backslash alpha^i$ or if and , then

$W(q;\backslash alpha^i)\backslash ge\; W(q\text{'};\backslash alpha^i)\; \backslash implies\; W(q;\backslash alpha^\{i\text{'}\})\backslash ge\; W(q\text{'};\backslash alpha^\{i\text{'}\})$

where W is the indirect utility function.

An important result extends the median voter theorem, which states that when voters have single peaked preferences, there is a majority-preferred candidate corresponding to the median voter's most preferred policy.{{Cite journal |last1=Bredereck |first1=Robert |last2=Chen |first2=Jiehua |last3=Woeginger |first3=Gerhard J. |date=October 2013 |title=A characterization of the single-crossing domain |url=http://link.springer.com/10.1007/s00355-012-0717-8 |journal=Social Choice and Welfare |language=en |volume=41 |issue=4 |pages=989–998 |doi=10.1007/s00355-012-0717-8 |s2cid=253845257 |issn=0176-1714}} With single-crossing preferences, the most preferred policy of the voter with the median value of $\backslash alpha^i$ is the Condorcet winner.{{cite book |last1=Persson |first1=Torsten |title=Political Economics: Explaining Economic Policy |last2=Tabellini |first2=Guido |publisher=MIT Press |year=2000 |isbn=9780262303668 |page=23}} In effect, this replaces the unidimensionality of policies with the unidimensionality of voter heterogeneity.{{Technical inline|date=November 2024}}{{cite journal |last1=Gans |first1=Joshua S. |last2=Smart |first2=Michael |date=February 1996 |title=Majority voting with single-crossing preferences |journal=Journal of Public Economics |volume=59 |issue=2 |pages=219–237 |doi=10.1016/0047-2727(95)01503-5|doi-access=free }} In this context, the single-crossing condition is sometimes referred to as the Gans-Smart condition.{{Cite journal |last1=Haavio |first1=Markus |last2=Kotakorpi |first2=Kaisa |date=May 2011 |title=The political economy of sin taxes |url=https://linkinghub.elsevier.com/retrieve/pii/S0014292110000607 |journal=European Economic Review |language=en |volume=55 |issue=4 |pages=575–594 |doi=10.1016/j.euroecorev.2010.06.002|hdl=10138/16733 |s2cid=2604940 |hdl-access=free }}

In mechanism design, the single-crossing condition (often referred to as the Spence-Mirrlees property for Michael Spence and James Mirrlees, sometimes as the constant-sign assumption{{Cite book |last1=Laffont |first1=Jean-Jacques|last2=Martimort|first2=David|url=https://www.worldcat.org/oclc/47990008 |title=The theory of incentives : the principal-agent model |date=2002 |publisher=Princeton University Press |isbn=0-691-09183-8 |location=Princeton, N.J. |pages=53 |oclc=47990008}}) refers to the requirement that the isoutility curves for agents of different types cross only once.{{Cite book |last1=Laffont |first1=Jean-Jacques|last2=Martimort|first2=David|url=https://www.worldcat.org/oclc/47990008 |title=The theory of incentives : the principal-agent model |date=2002 |publisher=Princeton University Press |isbn=0-691-09183-8 |location=Princeton, N.J. |pages=35 |oclc=47990008}} This condition guarantees that the transfer in an incentive-compatible direct mechanism can be pinned down by the transfer of the lowest type. This condition is similar to another condition called strict increasing difference (SID).{{Cite journal |last=Frankel |first=Alexander |date=2014-01-01 |title=Aligned Delegation |url=https://pubs.aeaweb.org/doi/10.1257/aer.104.1.66 |journal=American Economic Review |language=en |volume=104 |issue=1 |pages=66–83 |doi=10.1257/aer.104.1.66 |issn=0002-8282}} Formally, suppose the agent has a utility function $V(q,\backslash theta)$, the SID says $\backslash forall\; q\_2>q\_1,\backslash theta\_2>\backslash theta\_1$ we have $V(q\_2,\backslash theta\_2)-V(q\_1,\backslash theta\_2)>V(q\_2,\backslash theta\_1)-V(q\_1,\backslash theta\_1)$. The Spence-Mirrlees Property is characterized by $\backslash frac\{\backslash partial^2V\}\{\backslash partial\backslash theta\backslash partial\; q\}(q,\backslash theta)>0$.

• Brouwer fixed-point theorem

{{Reflist|group=note}}

{{reflist}}

Category:Asymmetric information

Category:Fixed-point theorems

Category:Utility function types