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It is also common to discount the surplus of the cook or introduce the stopping time, thus making the problem more accessible: the cook prefers less loss now to the bigger gain in the future. However, it is hard to justify any choice of discount factor in practice because nowadays many auctions are run by robots and ar... |
1.2. Contribution of this paper distortion at the top upper bound for the surplus of the fisher is established. Namely, if the fisher commits to a certain strategy in advance, then the higher the type [MATH] of the cook, the bigger share of the total welfare [MATH] distributed in each round goes to the cook. |
Then, we propose a strategy such that if the fisher commits to it, then the optimal in expectation response of the cook will be to play naively , i.e. accept the deal if and only if [MATH] , thus this strategy is incentive-compatible . Furthermore, the revenue of the fisher in this strategy attains the upper bound disc... |
The proposed mechanism is credible (cf. ), the fisher exercises ‘‘the power to commit’’: the cook can verify that the fisher is using exactly this strategy, so there will be no mistrust. |
1.3. The objectives of the players Fix a strategy [MATH] of the fisher and a strategy [MATH] of the cook. In the spirit of Wald’s maxmin model, we denote |
[EQUATION] [EQUATION] and say that the fisher’s objective is to maximize [MATH] and the cook’s objective is to maximize [MATH] . Roughly speaking, they maximize the minimal average revenue and the minimal average surplus respectively. Since the total welfare is [MATH] one may think than in each round the fisher and the... |
strategy (we allow mixed strategies too) is any rule which maps the previous history of the game to a distribution from where the next price (or a decision to accept or reject) is drawn. |
2. Observables and Spence-Mirrlees property Fix any strategy [MATH] of the fisher. Suppose that the cook is of the type [MATH] and the cook knows [MATH] in advance and chooses a strategy [MATH] . Let them play and write the history of all the moves, i.e. [MATH] . Taking the physical point of view we may try to extract ... |
[EQUATION] Again, we would like to take the limit of the average frequency of accepted deals. Since it may not exist, we consider [MATH] instead. |
Lemma 1 Suppose that [MATH] . Then, for each [MATH] , for [MATH] big enough we have [EQUATION] where we take the minimum by all strategies [MATH] of the cook. In other words, the fisher can not guarantee himself more than the share [MATH] of the total welfare for all sufficiently large [MATH] |
Proof. For [MATH] by playing [MATH] the cook of type [MATH] has [EQUATION] This gives [EQUATION] [EQUATION] for [MATH] big enough, which finishes the proof. |
This proof amounts to the fact that by pretending of being of the type [MATH] the cook takes roughly at least [MATH] share of the total welfare in each round. |
Let the cook of type [MATH] play a strategy [MATH] against [MATH] . It is natural to assume that the revenue of the cook of type [MATH] using strategy [MATH] is at least that of playing the strategy [MATH] with [MATH] , because otherwise we may set [MATH] |
Using the history of playing [MATH] against [MATH] we define [MATH] for all [MATH] as [EQUATION] Lemma 2 Under the above assumptions, for each [MATH] we have |
[EQUATION] Proof. If the cook’s type is [MATH] and he plays the strategy [MATH] , his average surplus is at least [EQUATION] which must be at most [MATH] |
Morally, this estimate tells us that the derivative of [MATH] (if it exists) is at least [MATH] . Also, this is the old good Spence-Mirrlees property. |
It is not difficult to show that if [MATH] locally decreases with [MATH] then is not possible that both fisher and cook are better when [MATH] increases. Therefore it is natural to assume that [MATH] grows with [MATH] . Additionally, one is likely to expect that the higher the type [MATH] of the buyer, the more frequen... |
2.1. Distortion at the top It directly follows from Lemma that the following property holds. Corollary 1 If [MATH] is non-decreasing function on [MATH] with [MATH] then |
[EQUATION] i.e the fraction that the fisher gets out of the total welfare [MATH] tends to zero. Note that using an arbitrary strategy [MATH] of the fisher and the responses [MATH] of the cook we constructed a function [MATH] , the observable which measures the proportion of accepted deals. Now we switch our attention t... |
3. A new mechanism Our mechanism is introduced after the following informal motivation. 3.1. Commitment Let [MATH] be the following naive strategy for the cook: accept the price [MATH] if [MATH] , refuse otherwise. For the cook of type [MATH] , playing [MATH] is not always optimal: the fisher quickly determines [MATH] ... |
On the other hand, imagine that the cook makes a commitment : he tells to the fisher that he is going to play [MATH] regardless the actions of the fisher. If the cook sticks to this strategy (fortunately for the fisher in many cases the cook has no such a commitment power), then the best response for the fisher is to a... |
If the fisher has a strategy, it makes sense to reveal it. Indeed, otherwise a strategic cook tricks the fisher and is afraid to buy even for a small [MATH] because this gives the fisher an information about the cook’s type, and the fisher will use this information in an unknown way. It is proven that the absence of a ... |
3.2. Strategy background ideas, fisher’s guess of the cook’s type A strategy of the fisher (choosing the price [MATH] after the round [MATH] ) can depend on the history of proposed prices [MATH] and the responses [MATH] of the cook in all rounds [MATH] preceding a given round. It seems reasonable for the fisher to squa... |
A good strategy [MATH] of the fisher has the following qualities: type adaptation [MATH] should depend on [MATH] and adapt to it (one may imagine that the true type [MATH] of the cook slowly changes over time an it would be good if the fisher’s strategy adapts to this change); |
rewarding , to incentivize the cook, the surplus of the cook should be monotone with respect to [MATH] , and even more: since the cook can pretend that his type is lower than it is, the growth of the cook’s surplus when the fisher’s estimate [MATH] increases must include what the cook can obtain by pretending that he i... |
type confirmation , the fisher should not incentivise the cook to pretend that his type is higher than it is, because stimulating to lie is not good by itself and because this again can crash the market and impose additional revenue loss for the fisher. |
To disentangle all these features we take each of them to the extreme and propose each of them with some probability (because mixed strategies frequently work better in the context of learning or truthful mechanism design, cf. |
). Start with rewarding : the fisher will sometimes propose [MATH] , i.e. will give the object to the cook for free. To implement type adaptation it is enough to propose a price higher than the current estimate [MATH] , for example, [MATH] is drawn from a uniform distribution on [MATH] (or any other distribution with t... |
3.3. The mechanism, formally The fisher knows that [MATH] (the mechanism can be easily adapted to the case when the fisher knows for sure that [MATH] belongs to a fixed interval). The fisher fixes any increasing function [MATH] Let [MATH] be the solution of the following differential equation: [MATH] . The fisher commi... |
In each round the fisher has an estimate [MATH] of [MATH] [MATH] ). The fisher plays the following mixed strategy [MATH] randomly choosing between three types of prices in this round: |
(with probability [MATH] ) the fisher plays type adaptation : the price [MATH] is uniformly drawn from [MATH] , accepting this plice the cook achieves that [MATH] . If [MATH] is accepted, then the fisher sets [MATH] . If [MATH] is refused, then [MATH] |
(with probability [MATH] ) the fisher plays rewarding : the price [MATH] , which is profitable for the cook, and the probability of this price appearing grows with [MATH] thereby stimulating the cook to reveal his type. |
(with probability [MATH] ) the fisher plays type confirmation : the price [MATH] , which is assumed to give no surplus to the cook, but refusing it incurs lowering the estimate of the type of the cook. If [MATH] is accepted, then [MATH] . If [MATH] is refused, then we should set [MATH] . This can be done in many ways, ... |
[EQUATION] The fisher sets [MATH] . The intuition for this formula is as follows: it is natural to define [MATH] (the empirical guess of [MATH] ) as the average between the maximal accepted price and the minimal rejected price. But, depending on the strategy of the cook, the latter can be less than the former. So we re... |
While the fisher plays [MATH] , the average surplus of the cook of type [MATH] , playing [MATH] is [EQUATION] where the last inequality follows from the definition of [MATH] if [MATH] , and from the convexity of [MATH] when [MATH] . This kind of inequality is common in Revenue Equivalence type theorems. |
From this we derive the following theorem. Theorem 1 For any strategy [MATH] of the cook we have [EQUATION] where [MATH] stands for the expectation (recall that the strategy [MATH] is not pure). |
Proof. The cook can manipulate [MATH] . The incentives are as follows: 1) If [MATH] then for the cook it is profitable because the cook gains additionally on accepting type adaptation price; 2) If [MATH] , then for the cook it is profitable because the reward price [MATH] appears more frequently. |
By the mechanism, to sustain [MATH] the cook has to always reject prices [MATH] , i.e. this strategy can not be more profitable than [MATH] . In order to make [MATH] bigger than [MATH] , the cook should accept type adaptation prices (which are rare, so the process will take a long time) and type confirmation prices, bo... |
To summarise, due to the strategy of the fisher, in order to make the fisher believe that [MATH] is [MATH] the cook should, in fact, play a strategy [MATH] for a substantial time, which is not profitable on average by construction. |
Surely, a risk loving cook can play [MATH] with [MATH] and this can be more profitable than playing [MATH] on short sequences of rounds. |
4. Discussion The main idea of this paper is to put the situation upside down: for a third party observing the game the most salient number which can be extracted from the game is the proportion [MATH] of accepted deals. It is a function [MATH] on the type [MATH] of the cook. Then to construct the mechanism we use this... |
As far as we know the proposed mechanism is novel (and very simple). Another distribution for mixed strategies may be considered in the fisher’s strategy, but they all should have an atom at [MATH] : indeed, if the surplus is growing with [MATH] , the cook has an incentive to pretend that his type is higher than it is.... |
) one can give the following metaphor: 1) an employee should have opportunities to show that she is more capable that her status suggests, 2) higher the status, more free benefits (ratchet effect), 3) there should be always work to do, to confirm her high status. |
The idea that in order to reveal the true type of the cook, the fisher must give him the substantial part of the total welfare (almost [MATH] of welfare when the cook’s type is huge) is not new and can be traced back to |
Note another advantage of the proposed mechanism: if the valuation [MATH] of the cook changes, the algorithm can be easily adapted – for example, we may set the rule for [MATH] taking into account only the last [MATH] deals. Similar statements can be found in |
(Section 6), but under a different abstract disguise, in another context, and without a concrete mechanism. The mechanism is credible: by playing enough time in a stable position (meaning that [MATH] does not change) the cook can calculate the frequency of reward, adaptation, and confirmation prices, and compare them w... |
4.1. How the fisher chooses [MATH] Indeed, instead of saying that the fisher knows the distribution of [MATH] we need to choose [MATH] , which, again, means to make a guess about distribution. A subtle difference is that it is hard to guess the distribution if we suspect that (surely, with small probability, but what i... |
4.2. Further questions How do real people behave playing this game? (cf. ) Let us say, two participants play 20 rounds, and it is known that [MATH] belongs to a given interval, but the distribution of [MATH] is not known to the fisher role player. The players should be paid: e.g. a fixed amount of money is equal to [MA... |
A multi-person game may be considered in the spirit of : let the fisher face several cooks (of a priori different types), the game be infinite, and there be no discounting. It seems that for any mechanism there will be a distortion at the top estimate, but only for the cook of the highest type, whose presence coerces a... |
# Source: arxiv 1806.02662 # Title: Almost Commutative Q-algebras and Derived brackets # Sections: all # Downloaded: 2026-03-03T02:41:46.350822+00:00 |
Almost commutative [MATH] -algebras and derived brackets Abstract. We introduce the notion of almost commutative Q-algebras and demonstrate how the derived bracket formalism of Kosmann-Schwarzbach generalises to this setting. In particular, we construct ‘almost commutative Lie algebroids’ following Vaĭntrob’s Q-manifol... |
Keywords: noncommutative geometry; almost commutative algebras; Lie algebroids; Q-manifolds MSC 2010: Primary 81R60 ; 46L87; 17B75 |
Secondary 53D17; 58A50 1. Introduction When quantum effects of the gravitational field become relevant, it is expected that the geometry of space-time will depart from its classical nature. From rather general arguments, it is likely that space-time at some level will no longer be a Riemannian manifold, but instead, ta... |
. From a mathematical perspective, noncommutative geometry is the natural progression of space-algebra duality and sheds light on what one could possibly mean by ‘geometry’. There are many approaches to noncommutative geometry and here we broach the subject by employing a very mild form of noncommutativity. |
Almost commutative algebras, also known as [MATH] -commutative algebras, have been studied since the 1980s following the work of Rittenberg & Wyler |
and Scheunert on generalised Lie algebras. Loosely, we have an algebra that is commutative up to some “commutation factor”. It was shown by Bongaarts & Pijls |
that almost commutative algebras offer a convenient framework to develop a very workable form of noncommutative differential geometry. A natural example of such an algebra is the [MATH] -commutative algebra of global functions on a supermanifold where the commutation factor is simply a plus or minus sign. We remark tha... |
. In particular, almost commutative algebras are ‘close enough’ to commutative and supercommutative algebras to allow many of the constructions found in differential geometry to be directly generalised. Using almost commutative algebras one can largely mimic classical differential geometry following the derivation base... |
. Many of the subtleties of working with “fully noncommutative” algebras disappear when using almost commutative algebras. For example, [MATH] -derivations of an almost commutative algebra form a module over the whole of the algebra. Generically one can develop lots of geometry with minimal fuss by employing almost com... |
. Specifically, one does not need to reach for the [MATH] -algebraic approach to noncommutative geometry as pioneered by Connes in order to build “mildly” noncommutative spaces. Moreover, interesting and well-known examples of such algebras exist such quantum hyperplanes, super-versions thereof, noncommutative tori, an... |
In this paper, we modify the understanding of Lie algebroids due to Vaĭntrob to the setting of almost commutative algebras. We will work with very particular almost commutative algebras that carry an additional [MATH] -grading, which we will refer to as weight . The assignment of weight is quite independent of the comm... |
. This idea has since been refined and applied by many authors (see for example ). These algebras will come equipped with a homological derivation (to be defined carefully in the main sections), and we will show that the derived bracket formalism of Kosmann-Schwarzbach |
(also see Voronov ) allows one to construct a ‘shifted’ [MATH] -Lie bracket on the space of particular derivations of the almost commutative algebra. All these constructions parallel the classical understanding of Lie algebroids in terms of graded supermanifolds equipped with a homological vector field. Indeed, up to a... |
While the notion of Lie–Rinehart pair as an algebraic generalisation of a Lie algebroid has been well studied, our motivation for this work was to understand if the picture of Lie algebroids due to Vaĭntrob has a natural generalisation in the noncommutative world. While we offer no perspective on the full noncommutativ... |
, are prevalent throughout differential geometry, geometric mechanics, and geometric approaches to field theory in many guises. Our general reference for Lie algebroids is the book by Mackenzie |
. Moreover, understanding Lie algebroids and related objects in terms of supergeometry has slowly been gaining acceptance. It seems natural and fitting that a similar understanding of ‘algebroids’ in the noncommutative setting be reached. Furthermore, it is well known that the categories of Poisson manifolds and Lie al... |
. Given that the quasi-classical limit of a ‘noncommutative space-time’ is expected to be related to a Poisson algebra, understanding ‘noncommutative Lie algebroids’ in all their formulations could be a fruitful avenue of exploration. |
Arrangement: In Section we review the notion of an almost commutative algebra. It is here that we introduce the concept of homological [MATH] -derivations and almost commutative Q-algebras. In Section we present the notion of an almost commutative non-negatively graded algebra. In Section we study almost commutative Q-... |
2. Almost Commutative Algebras We draw heavily on the work of Bongaarts & Pijls , Ciupală and Ngakeu in this section. We will present no proofs for the statements in this section as they follow directly from calculations or can be found in the literature previously cited. Let [MATH] be a discrete abelian group, which w... |
[EQUATION] for all [MATH] and [MATH] . One can deduce that these conditions imply that [MATH] [MATH] and that [MATH] , for all for all [MATH] and [MATH] . Moreover, we also have that [MATH] . The mapping [MATH] is a particular [MATH] -cocycle on the group [MATH] and is often referred to a commutation factor . For the r... |
Two cocycles [MATH] and [MATH] on [MATH] are said to be equivalent cocycles if and only of there exists an element [MATH] , i.e., an element of the automorphism group of [MATH] , such that [MATH] for all [MATH] and [MATH] |
Example 2.1 If [MATH] , then the trivial cocycle is given by [MATH] and the standard cocycle is given by [MATH] . The standard cocycle is, of course, just the standard sign factor. |
Example 2.2 If [MATH] and [MATH] , then [MATH] is a cocycle. As a [MATH] -vector space, we have the natural definition of [MATH] -parity for [MATH] -homogeneous elements of [MATH] |
[EQUATION] We then say that a homogeneous element [MATH] is even if [MATH] and odd if [MATH] . For homogeneous elements [MATH] and [MATH] , we define the [MATH] -commutator as |
[EQUATION] Extension to non-homogeneous elements is via linearity. Whenever we write an expression for homogeneous elements, we will understand that extension to non-homogeneous elements is via linearity, although we will not state this explicitly. The reader can directly verify the following properties: |
[EQUATION] A direct calculation yields the [MATH] -Jacobi identity [EQUATION] The properties of the [MATH] -commutator naturally suggest the definition of a [MATH] -Lie algebra , a notion which can be traced at least to Rittenberg & Wyler |
, Scheunert and Mosolova . Such generalised Lie algebras are also known as Lie colour algebras . We will later require the notion of a ‘shifted’ or ‘anti’ Lie algebra. We allow the bracket to carry a non-trivial [MATH] -degree, however, we will insist on ‘oddness’. The reader should keep in mind the classical Schouten–... |
Definition 2.3 Let [MATH] be a [MATH] -graded vector space and let [MATH] be a cocycle. Then [MATH] is a [MATH] -Loday-Leibniz antialgebra of [MATH] -degree [MATH] if there exists a bilinear map, which we refer to as a [MATH] -antibracket |
[EQUATION] such that (i) [MATH] (ii) [MATH] (iii) [MATH] for all [MATH] and [MATH] If in addition to the above bracket has the following [MATH] -skewsymmetry property |
(iv) [MATH] then we speak of a [MATH] -Lie antialgebra of [MATH] -degree [MATH] In the above definition, the [MATH] -Jacobi identity we have written in so-called Loday–Leibniz form (cf. Loday |
). This definition has the natural interpretation of [MATH] being a [MATH] -derivation of [MATH] -degree [MATH] of the ‘bracket algebra’. Note that this interpretation is quite independent of the [MATH] -skewsymmetry. |
Remark 2.4 Our nomenclature ‘Lie antialgebra’ and ‘antibracket’ has been hijacked from the BV-BRST formalism in physics. Mathematically, ‘anti’ signifies a shift in the grading as compared to the standard case of Lie superalgebras. In the current context, ‘anti’ signifies that [MATH] and not [MATH] |
With these notions in place, we have the following definitions that will be central to the rest of this paper. Definition 2.5 Let [MATH] be a [MATH] -graded algebra equipped with a given cocycle [MATH] . Then [MATH] is said to be a [MATH] -commutative algebra or an almost commutative algebra if and only if [MATH] for a... |
Note that in the above we fix a cocycle in the definition of an almost commutative algebra. That is, we will not consider group automorphisms and equivalent cocycles in any of the constructions in this paper. Moreover, associativity and “almost commutativity” force the cocycle to have the properties postulated - this i... |
Remark 2.6 The term [MATH] -graded commutative algebras was used in and the term colour commutative algebras was used in for what we call almost commutative algebras. |
In particular, for odd elements of an almost commutative algebra, it is clear that [MATH] . Thus, we have nilpotent elements to contend with. However, the assignment of [MATH] -parity does not determine the commutation rule [MATH] |
Definition 2.7 Let [MATH] and [MATH] be almost commutative algebras with respect to some fixed abelian group [MATH] and cocycle [MATH] . Then a morphism of almost commutative algebras is a [MATH] -linear map [MATH] , such that |
(i) [MATH] , and (ii) [MATH] for all [MATH] and [MATH] Note that we consider only degree zero morphisms, that is, morphisms do not change the [MATH] -degree of elements. This requirement is very natural from the perspective of constructing geometries, but from an algebraic perspective, this can be relaxed. The reader c... |
Example 2.8 (Commutative algebras) Any commutative algebra can be considered as an almost commutative algebra by taking [MATH] as the trivial group and the trivial cocycle. As a specific example, the algebra of global functions on a smooth manifold is a commutative algebra. |
Example 2.9 (Supercommutative algebras) Any supercommutative algebra can be considered as an almost commutative algebra by taking [MATH] and the cocycle to be [MATH] . As a specific example, the algebra of global functions of a supermanifold is a supercommutative algebra. |
Example 2.10 (Quaternions The quaternionic algebra [MATH] is [MATH] and [MATH] , subject to the relations [EQUATION] The quaternionic algebra is an almost commutative algebra with [MATH] , where the generators are assigned the grading |
[EQUATION] The relevant cocycle is given by [EQUATION] Example 2.11 [MATH] -commutative algebras [MATH] -graded, [MATH] -commutative algebra ( [MATH] ) can be considered as an almost commutative algebra by taking [MATH] [MATH] -times) and the cocycle to be [MATH] , where [MATH] is the standard scalar product on [MATH] ... |
). Remark 2.12 It has been long known that [MATH] -gradings play a fundamental rôle in physics as soon as fermions are taken into account. Far less well known is that [MATH] -gradings [MATH] naturally appear in the context of Green’s parastatistics, see for example Drühl, Haag & Roberts |
Example 2.13 (2-d quantum plane) The quantum plane is described by the algebra [MATH] , which is [MATH] and [MATH] , subject to the relation |
[EQUATION] where [MATH] is fixed and non-zero. The algebra [MATH] is naturally [MATH] -graded, [EQUATION] where [MATH] is the one-dimensional vector space spanned by products of the form [MATH] . Note that [MATH] if either [MATH] or [MATH] are negative. It is not difficult to see that the relation on the generators tra... |
[EQUATION] Clearly, we have an almost commutative algebra by defining the cocycle as [EQUATION] We just remark that the [MATH] -dimensional quantum hyperplane can similarly be defined and understood in the context of almost commutative algebras. Similarly, one can consider ‘super’ versions of quantum hyperplanes follow... |
. Such algebras are often referred to as quasi-commutative algebras Example 2.14 (Noncommutative 2-torus) The algebra of functions on a noncommutative 2-torus denoted as [MATH] , is the algebra [MATH] and [MATH] , subject to the relation |
[EQUATION] with [MATH] . Much like the example of the 2-d quantum plane the algebra [MATH] is naturally [MATH] -graded [EQUATION] |
where [MATH] is the one-dimensional vector space spanned by products of the form [MATH] . Again in almost the same way as the 2-d quantum plane, we naturally have an almost commutative algebra by defining the cocycle as |
[EQUATION] Noncommutative [MATH] -tori can similarly be defined and these are also further examples of almost commutative algebras. |
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