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Applying this in ( ) we obtain that the maximal possible dimension of an abelian subspace in [MATH] can only be equal to one of the following six sums: |
[EQUATION] To find out whether there is an abelian subspace of half the dimension in [MATH] is thus equivalent to finding out whether subtracting from the dimension of [MATH] , i. e. from [MATH] , one of these sums doubled gives zero, i. e. whether one of the sums |
[EQUATION] is zero. Simplifying, we obtain respectively [EQUATION] Rewriting this further as [EQUATION] and taking ( ) into account this can be rewritten as |
[EQUATION] Let us now assume that our nilpotent is strictly odd, which in terms of the corresponding partition means that [MATH] (here as before [MATH] is the largest nonzero part of the partition). This then implies that all multiplicities [MATH] are nonzero. Thus to obtain an abelian subspace of half the dimension in... |
[EQUATION] We now make the following observations, according to the parity of [MATH] if [MATH] is odd, then the cases in the first column are not realizable, since they require that the partition has no even parts, while by strict oddity both [MATH] and [MATH] must be nonzero; |
if [MATH] is even, the cases in the second column are not realizable by exactly the same reason. Taking this into account, we are left with the following cases: for [MATH] even, |
[EQUATION] and for [MATH] odd, [EQUATION] Let us also observe the following: for [MATH] even, the first case is subsumed by the third one; |
for [MATH] even, the third case is subsumed by the second one for type D; for [MATH] odd, the subcase [MATH] of the third case is subsumed by the first one for type B, and by the second one for type D. |
Taking all of the above into account gives the partitions as described. Remark 3.5 One can formulate the theorem more understandably as follows: in case of type C, there is exactly one parity change along the partition, while in cases B or D there might be either one or two parity changes, but if there are two parity c... |
We now turn to the not necessarily strictly odd nilpotent orbits, using strictly odd reduction from 2.4 . For classical types, its reformulation in terms of partitions is as follows. |
Lemma 3.6 Let [MATH] be a simple Lie algebra of classical type, and let [MATH] be a nilpotent element of [MATH] corresponding to the partition [MATH] , with [MATH] such that [MATH] and [MATH] are of opposite parity while all the larger parts [MATH] (those with [MATH] ) are of the same parity. |
Then the partition [MATH] defines a strictly odd nilpotent in a Lie algebra of the same type, and corresponds to the strictly odd reduction of [MATH] , as defined in 2.4 |
Proof. Let us begin by noting that the modified partition is indeed suitable for the same type: if this requires that all parts of the same parity as [MATH] have even multiplicity, then we have not touched them; while if this requires that all parts of the same parity as [MATH] are even, then [MATH] and all larger part... |
Now following the correspondence between partitions and weighted Dynkin diagrams described above it is easy to see that passing from the original partition to the one modified as described corresponds to the following modification of the weighted Dynkin diagram: one removes all nodes (and weights) from the left until n... |
But this precisely means to leave the connected component of the weighted Dynkin diagram that contains nonzero weights, as described in 2.2 above, so that we indeed obtain the strictly odd reduction of [MATH] |
Corollary 3.7 Given a nilpotent in a simple Lie algebra [MATH] of classical type [MATH] [MATH] or [MATH] , there is an abelian subspace of half dimension in [MATH] if and only if the partition corresponding to the nilpotent satisfies the following conditions: |
type [MATH] there is no more than one parity change along the partition; types [MATH] and [MATH] there are no more than two parity changes and, if there is at least one parity change then |
if the largest part of the partition is even, then there is only one parity change, and in the [MATH] case moreover it must be the unique even part and must have multiplicity [MATH] |
if there are two parity changes, then the largest part of the partition is odd, there is a unique even part, it has multiplicity [MATH] , and all smaller parts are equal to [MATH] |
Thus, abelian subspaces of half dimension in [MATH] occur precisely for nilpotents corresponding to partitions of one of the following kind (with [MATH] throughout): |
any type: [MATH] type C or D: [MATH] type B or D: [MATH] type B: [MATH] Proof. This follows from 3.6 . Indeed the latter shows that [MATH] for a nilpotent [MATH] corresponding to some partition has an abelian subspace of half dimension if and only if [MATH] , as described in 2.3 , has such a subspace; and this happens ... |
It remains to note that a partition is of the kind indicated if and only if the partition obtained from it as in 3.6 satisfies conditions of 3.3 |
Computations It thus remains to find out which of the strictly odd nilpotent orbits in simple Lie algebras of exceptional type do possess an abelian subspace of half dimension in degree 1. |
For that, we used the computer algebra system GAP . In the package SLA by Willem A. de Graaf included in this system one can compute with nilpotent orbits of arbitrary semisimple Lie algebras. In particular, one obtains canonical bases consisting of root vectors for the homogeneous subspaces of all degrees in the gradi... |
Using 2.1 , we can determine abelian subspaces in [MATH] as follows. Let [MATH] be the basis of [MATH] made from positive root vectors. Let us construct a graph with the set of vertices [MATH] , where two vertices [MATH] and [MATH] are connected with an edge if and only if they do not commute, that is, if and only if [... |
Clearly this is equivalent to the corresponding graph having an independent set of cardinality [MATH] — that is, a subset consisting of [MATH] vertices such that no two of these vertices are connected by an edge. Hence describing all possible dimensions of abelian subspaces in [MATH] reduces to listing all possible car... |
There is another package GRAPE by Leonard H. Soicher in GAP which can be used to list all independent sets in a finite graph. Using this package we determine independent sets of maximal possible cardinality in the graph corresponding to the nilpotent orbit. |
The results are given in the tables below. A GAP code for computing maximal dimensions of abelian subspaces in [MATH] for arbitrary semisimple Lie algebras is available at |
. In fact the program can list all subsets of any given cardinality of pairwise commuting elements in the root vector basis. As an illustration, here are two cases for E |
Examples 4.1 The nilpotent orbit with the weighted Dynkin diagram [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
has [MATH] of dimension [MATH] The corresponding graph with 14 vertices and edges connecting vertices corresponding to non-commuting root vectors in [MATH] looks as follows: |
This graph has independent sets with [MATH] vertices, e. g. [MATH] , but any subset on more than [MATH] vertices contains a pair of vertices connected with an edge, thus for this nilpotent orbit maximal dimension of an abelian subspace is equal to [MATH] |
Another orbit in E , with the diagram [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] , has [MATH] of dimension [MATH] corresponding to the graph |
with 10 vertices. It is easy to find in this graph an independent subset with five elements – e. g. [MATH] Thus the orbit in the first example does not possess an abelian subspace of half dimension in [MATH] , while that in the second one does. |
Tables Table G2s Strictly odd nilpotent orbits in G , all with half-abelian [MATH] Name Diagram [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
Table F4s Strictly odd nilpotent orbits in F with half-abelian [MATH] without half-abelian [MATH] Name Diagram [MATH] Name Diagram |
[MATH] (largest dimension of an abelian subspace) [MATH] [MATH] [MATH] [MATH] 14 [MATH] [MATH] [MATH] [MATH] [MATH] 8 (2) [MATH] |
[MATH] [MATH] [MATH] [MATH] 8 (3) Table E6s Strictly odd nilpotent orbits in E with half-abelian [MATH] without half-abelian [MATH] |
Name Diagram [MATH] Name Diagram [MATH] (largest dimension of an abelian subspace) [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
10 [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Table E7s Strictly odd nilpotent orbits in E |
with half-abelian [MATH] without half-abelian [MATH] Name Diagram [MATH] Name Diagram [MATH] (largest dimension of an abelian subspace) |
16 Table E8s Strictly odd nilpotent orbits in E with half-abelian [MATH] without half-abelian [MATH] Name Diagram [MATH] Name Diagram |
[MATH] (largest dimension of an abelian subspace) [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
(Non-strictly) odd nilpotent orbits in F , all with half-abelian [MATH] Name Diagram Strictly odd piece [MATH] [MATH] [MATH] [MATH] |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Table E6o (Non-strictly) odd nilpotent orbits in E , all with half-abelian [MATH] Name |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Table E7o (Non-strictly) odd nilpotent orbits in E |
with half-abelian [MATH] without half-abelian [MATH] Name Diagram Strictly odd piece Name Diagram Strictly odd piece [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
[MATH] Table E8o (Non-strictly) odd nilpotent orbits in E with half-abelian [MATH] without half-abelian [MATH] Name Diagram Strictly odd piece |
The authors are grateful to the referee for highly professional work, including simplifications of proofs and improvements of exposition. |
The third named author wishes to thank Daniele Valeri for discussions on generalization of Miura maps, constructed in The second named author gratefully acknowledges help of the user marmot from tex.stackexchange.com in producing the code that was used to highlight the strictly odd pieces of weighted Dynkin diagrams in... |
# Source: arxiv 1806.00898 # Title: Competitive pricing despite search costs if lower price signals quality # Sections: all # Downloaded: 2026-03-02T09:24:02.800745+00:00 |
Competitive pricing despite search costs if lower price signals quality Abstract I show that firms price almost competitively and consumers can infer product quality from prices in markets where firms differ in quality and production cost, and learning prices is costly. Bankruptcy risk or regulation links higher qualit... |
Keywords: Price signalling, Diamond paradox, price dispersion, incomplete information, price war. JEL classification: D82, C72, D41. |
In many markets, looking up prices is costly for consumers. Even in online markets, the few moments it takes to check a (second) website constitutes a positive cost. Firms usually have private information about their cost and quality. This paper shows that with negatively related private cost and quality, firms set a p... |
Amazon has greater variety, faster delivery and a lower cost per package delivered than smaller online sellers. Regulation can cause a low-quality firm to have a higher cost. If a low-quality firm’s product is more likely to be faulty and a regulator punishes firms for faulty products, then a low-quality firm has a gre... |
Optimal allocation of managerial talent (or some other resource) between cost reduction and quality improvement also links lower cost to higher quality. Improving quality is subject to moral hazard, because consumers pay based on the quality they expect, not the quality that the firm chooses. Cost reduction benefits th... |
In the markets studied in this paper, there are at least two firms, each of which draws an independent type, either good or bad . The good type has lower marginal cost and higher quality than the bad. Each firm knows its own type. The consumers and the other firms only have a common prior over a firm’s type. First the ... |
In equilibrium, prices are close to competitive due to a race to the bottom that consists of two forces. One is downward price signalling, i.e. the high-quality firm reduces price to distinguish itself from the low-quality firm and attract greater demand. The second force is that a low-quality firm cuts price to deter ... |
The race to the bottom ends when the high-quality firm prices at the marginal cost of the low-quality firm. The low-quality firm’s price is its marginal cost plus the minimal monetary unit—the same as under complete-information Bertrand competition between two low-quality firms when the consumers have zero learning cos... |
Competition with privately known cost and quality also contrasts with monopoly under private information, and with competition when quality is observed together with the price. In monopoly, the good type still signals its quality by reducing its price to a level less profitable for the bad type than the bad type’s mono... |
In competition when paying a learning cost leads to observing both price and quality, the low-quality firms are still in a Bertrand-like situation and compete to a low price. However, a high-quality firm has no incentive to reduce price to signal, because customers stay at a high-quality firm even at a high price. The ... |
Empirical evidence for the model can be found in the car industry, namely that high-quality cars have a lower price and production cost. According to Vasilash ( 1997 , the assembly cost of more reliable cars is smaller, controlling for vehicle category, e.g. subcompact, compact, etc. The rank correlation between the pr... |
and the CarMD index of repair incidents (higher index means more breakdowns) is positive, but statistically insignificant in the sample of 40 cars that belong to both the top 288 new cars by number sold in the US in 2015 |
and the 100 most reliable in 2015 according to CarMD. The average cost per repair is also larger for more expensive cars according to CarMD, but this is less surprising, because parts for more expensive vehicles cost more. The cheapest cars to maintain according to YourMechanic are those of East Asian manufacturers, wi... |
Given the above, it is not surprising that the per-car profits are higher for Toyota than for Detroit’s Big 3 automakers (Wayland, 2015 . Competition has resulted in approximately zero profit for the higher-cost manufacturers. For example, the US government had to bail out Detroit’s Big 3 carmakers during the 2008 fina... |
Literature The foremost paper on costly learning of prices is Diamond ( 1971 , where competing firms set the monopoly price. A monopoly price or above is also found in Diamond ( 1987 ); Axell ( 1977 ); Reinganum ( 1979 ); Klemperer ( 1987 and Garcia et al. ( 2017 A number of solutions to the Diamond paradox have been p... |
The current paper does not rely on zero learning cost, multiple free price observations, taste shocks or repeat purchases. To the author’s knowledge, this work is the first to combine signalling and consumer search costs. The informative price difference between firm types endogenously gives consumers the incentive to ... |
Downward price signalling by a single firm has been studied in Shieh ( 1993 . A similar idea is in Simester ( 1995 , where multiproduct firms (whose prices for all products are positively correlated) signal by a low price on one product. In Rhodes ( 2015 , a multiproduct monopolist stocking more products (better for th... |
The receivers of the price signal are the consumers in this paper, which differs from limit pricing (as in Milgrom and Roberts ( 1982 and the literature following) where the receivers are potential entrants. |
The next section sets up the model. Section constructs an equilibrium with near-competitive pricing in a market with consumer search costs. and shows that this equilibrium is the unique one that survives the Intuitive Criterion of Cho and Kreps ( 1987 . The robustness of the results to relaxing various assumptions is d... |
Price competition with costly learning of prices There are two firms indexed by [MATH] , each with a type [MATH] (good and bad, respectively). Each firm knows its own type, but not that of the other. Types are i.i.d. with [MATH] . There is a continuum of consumers of mass [MATH] with types [MATH] distributed according ... |
The timeline of the game is as follows. 1. Nature draws independent types for firms and consumers, and assigns half the consumers to one firm, half to the other, independently of types. Each player observes his own type, but not the types of the others. |
2. Firms simultaneously set prices. 3. Each consumer observes the price of his assigned firm and chooses either to buy from this firm, learn the price of the other firm, or leave the market. |
4. Each consumer who chose to learn observes both firms’ prices and chooses either to buy from his assigned firm, buy from the other firm, or leave the market. |
A type [MATH] firm has marginal cost [MATH] normalised to [MATH] , and type [MATH] has [MATH] . The quality of a type [MATH] firm is higher. Specifically, a type [MATH] consumer values firm type [MATH] ’s product at [MATH] and [MATH] ’s product at [MATH] , with [MATH] [MATH] . To ensure that demand for [MATH] ’s good i... |
After the firms’ cost and quality are determined, the firms simultaneously set prices [MATH] , where [MATH] is the smallest monetary unit. |
Assume [MATH] for some [MATH] (costs are measured in terms of the minimal monetary unit, and not all consumers buy at a price just below the bad type’s cost). Assume [MATH] . Prices above [MATH] are unavailable w.l.o.g., because no consumer buys at any [MATH] |
For a set [MATH] , denote the set of probability distributions on [MATH] by [MATH] A behavioural strategy of firm [MATH] is [MATH] , so [MATH] is the probability that type [MATH] of firm [MATH] puts on price [MATH] |
A consumer sees the price that his assigned firm sets and can learn the price of the other firm at cost [MATH] Define [MATH] . Assume that [MATH] , i.e. the learning cost is small relative to the prior probability of the good type firm and the valuation difference between consumer type [MATH] for a good type firm and c... |
After seeing the price of his assigned firm, a consumer decides whether to buy from this firm (denoted [MATH] ), learn the other firm’s price ( [MATH] ) or not buy at all ( [MATH] ). Upon learning the price of the other firm, the consumer decides whether to buy from firm [MATH] (denoted [MATH] ), firm [MATH] [MATH] ) o... |
A type [MATH] firm’s ex post payoff if mass [MATH] of consumers buy from it at price [MATH] is [MATH] Assume that the full-information monopoly profit function [MATH] of firm type [MATH] strictly increases in [MATH] on [MATH] , so that the full-information monopoly price [MATH] of [MATH] is strictly above [MATH] (this ... |
A consumer’s posterior belief about firm [MATH] after observing its price [MATH] and expecting the firm to choose strategy [MATH] is |
[EQUATION] whenever [MATH] , and arbitrary otherwise. The gain from trade that consumer type [MATH] expects from buying from firm [MATH] at price [MATH] is denoted [MATH] |
The solution concept used is perfect Bayesian equilibrium (PBE), hereafter simply called equilibrium. Later, a unique equilibrium is selected using the Intuitive Criterion of Cho and Kreps ( 1987 |
Definition 1 An equilibrium consists of [MATH] and [MATH] satisfying the following for [MATH] [MATH] [MATH] [MATH] (a) if [MATH] , then [MATH] , and if in addition [MATH] , then [MATH] |
(b) if [MATH] , then [MATH] (c) if [MATH] , then [MATH] (d) if [MATH] then [MATH] (e) if [MATH] then [MATH] (f) if [MATH] , then [MATH] , where |
[EQUATION] (g) if [MATH] or [MATH] , then [MATH] is derived from ( ). The equilibrium profit of type [MATH] of firm [MATH] is denoted [MATH] ; it equals [MATH] for any [MATH] s.t. [MATH] |
Some tie-breaking rules are built into the equilibrium definition, e.g. a consumer indifferent between [MATH] and [MATH] chooses [MATH] . The results remain substantially the same if the tie-breaking rules are modified, as discussed in Section The next section constructively proves equilibrium existence by guessing and... |
Equilibrium This section constructs an equilibrium in which consumers put probability one on a firm being the good type if the price is below the bad type’s cost, otherwise probability one on the bad type. The good type firm sets a price equal to the bad type’s cost. The bad type’s price is its cost plus [MATH] . A con... |
The formal definition of the guessed equilibrium is the following: 1. Beliefs: [MATH] and [MATH] for [MATH] 2. Each firm’s type [MATH] sets price [MATH] and type [MATH] sets [MATH] |
3. If [MATH] , then [MATH] and [MATH] 4. If [MATH] and [MATH] , then [MATH] If [MATH] and [MATH] , then [MATH] 5. If [MATH] , then [MATH] , and if in addition [MATH] , then [MATH] If [MATH] , then [MATH] |
Part 1 of the guessed equilibrium includes Definition 1(g) and also specifies beliefs at off-path prices. Beliefs are consistent with Bayes’ rule ( ). Part 2 specialises Definition 1(f) to the guessed equilibrium. Parts 3–5 are simply the rewriting of Definition 1(a)–(e). Appendix proves that no player can profitably d... |
The idea of the proof is as follows. Consumers are clearly best responding to their belief, which is consistent with firm strategies. The bad type does not price below [MATH] , because it is weakly dominated by [MATH] . If consumers at a bad type learn and the other firm is the good type, then all consumers leave the b... |
The guessed equilibrium already partly resolves the Diamond paradox, because its outcome differs from monopoly pricing and no search. Prices in the guessed equilibrium are close to competitive. Type [MATH] prices the same as under Bertrand competition between the [MATH] types with zero search cost and complete informat... |
The following lemma shows the monotonicity of equilibrium demand and prices. Given the ranking of the costs and qualities of the types, the results are intuitive—the lower-cost type [MATH] sets a lower price and the higher-quality type [MATH] receives higher demand. Based on Lemma , there cannot be two prices on which ... |
Lemma 1 In any equilibrium, if [MATH] and [MATH] , then [MATH] , and if in addition [MATH] , then [MATH] The proofs of this and subsequent results are in Appendix |
The next lemma shows that in any equilibrium satisfying the Intuitive Criterion in which not all consumers buy at price [MATH] and belief [MATH] , neither firm sets a price at which demand is zero. Both types of both firms make positive profit, and the types set different prices with positive probability. To state the ... |
[EQUATION] The function [MATH] is the ratio of demand at belief [MATH] to demand at belief [MATH] . This function only depends on the primitives [MATH] , so [MATH] only depends on exogenous parameters. The ratio of demands is strictly greater than [MATH] and continuous when the denominator is positive (as is the case w... |
Lemma 2 For any [MATH] [MATH] and [MATH] , in any equilibrium satisfying the Intuitive Criterion, we have [MATH] and there exists [MATH] s.t. [MATH] and [MATH] |
Lemma provides the first component of the race to the bottom, namely the good types separating (at least partially) from the bad by setting a lower price. The Intuitive Criterion drives the separation, because it eliminates belief threats at low prices, which would otherwise deter the good types from price-cutting. |
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