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The next lemma establishes a lower bound on the equilibrium price by showing that the good types price weakly above the cost of the bad type. |
Lemma 3 For any [MATH] and [MATH] , in any equilibrium satisfying the Intuitive Criterion, if [MATH] , then [MATH] The intuition for Lemma is that the firms’ good types are in a race to the top at prices in [MATH] Neither firm’s good type loses customers to the other firm when raising price slightly, because the small ... |
In the unique equilibrium surviving the Intuitive Criterion, each good type sets price [MATH] and each bad type [MATH] , as shown in the following Theorem. The proof provides the second component of the race to the bottom: a bad type reduces price to deter its customers from learning and to undercut the other firm’s ba... |
Theorem 4 For any [MATH] and [MATH] , in the unique equilibrium satisfying the Intuitive Criterion, [MATH] Theorem shows that the unique equilibrium outcome that satisfies the Intuitive Criterion is the guessed equilibrium from above. Prices are close to competitive and there is positive, but small price dispersion. Th... |
(Section discusses cases outside that range). The equilibrium in Theorem is distinct from signalling by a monopoly, because a bad type monopolist does not have an incentive to cut price when the good type’s price is low enough. This is because there is no competing firm for the customers to learn about and leave to. Th... |
Section 3.1 below contrasts Theorem with competition when the type is observed together with the price. The comparisons to monopoly and observed type show that the combination of signalling and multiple firms is necessary as well as sufficient to overcome the effect of the positive search cost. |
Bertrand competition under zero learning cost between two known bad or two known good types leads to equal profits (close to zero) for the firms and no price dispersion, unlike in the equilibrium in Theorem . Bertrand competition between a good and a bad firm yields zero demand for the bad firm, but positive demand and... |
If some consumers have zero and others positive learning cost, but there is no quality or cost uncertainty, then the firms mix over an interval of prices between the competitive and the monopoly price. The price distribution depends strongly on the density of the learning costs at zero, and whether there is an atom at ... |
With consumer taste shocks (horizontal differentiation of firms), there is no price dispersion, and for each firm, some consumers initially at it learn another firm’s price and leave. This differs from the current paper, which models vertical differentiation and shows that consumers initially at a good firm do not lear... |
Models of repeat purchases have many equilibria, some of which replicate the pricing patterns found in this paper. However, the markets described by repeated games with high discount factors differ from the markets studied in this paper, which involve infrequent buying (repair services, insurance, durable goods such as... |
The next section relaxes some of the assumptions made above. The equilibrium remains qualitatively similar, in particular the Diamond paradox is still resolved. |
Robustness Relaxing the assumption that the full-information monopoly price [MATH] of the good type is above the cost of the bad type, the equilibrium price of the good type is either [MATH] as above (if [MATH] ), or [MATH] . In the latter case, the only modification of the equilibrium in Section is that [MATH] sets pr... |
If the learning cost is large enough ( [MATH] ), then some customers initially at a bad type setting price [MATH] buy immediately instead of learning the other firm’s price. These customers are called captive The mass of captive customers depends on [MATH] . If this is large, then the bad type sets price [MATH] with po... |
If there is a distribution of learning costs with [MATH] and [MATH] , then the equilibrium outcome is unchanged. Learning costs strictly greater than [MATH] create captive consumers, as discussed above. |
Nonpositive learning costs for some consumers eliminate the Diamond paradox even without incomplete information, as the previous literature showed. In the current model, enough consumers with a nonpositive learning cost make the good types reduce price, but the positive probability of the other firm having a bad type e... |
If all consumers buy at price [MATH] and belief [MATH] (formally, [MATH] ), then there is no reason for a good type to reduce price below [MATH] to increase belief. Both firms pooling on [MATH] survives the Intuitive Criterion, because if belief at any [MATH] is set to [MATH] and the good type wants to deviate to [MATH... |
The results remain unchanged if the tie-breaking rule for [MATH] in Definition depends on the belief or the price, e.g. if [MATH] , then the customer buys from the firm with the greater [MATH] (or smaller [MATH] ) with probability [MATH] . The results also do not change if ties are always broken in favour of a particul... |
A small asymmetry between firms has a similar effect to asymmetric tie-breaking. Denote type [MATH] of firm [MATH] by [MATH] . If consumers slightly prefer [MATH] to [MATH] , other things equal (interpreted as [MATH] having lower quality), then [MATH] gets zero demand and profit at equal price to [MATH] , because consu... |
If the firms can set any price in [MATH] for some [MATH] (not constrained to a grid), then an equilibrium satisfying the Intuitive Criterion does not exist. The proofs of Lemmas still work, but in Theorem , the bad types Bertrand compete down to price [MATH] . Then belief at [MATH] is strictly lower than [MATH] , the b... |
Having more than two firms only strengthens competition. Because the bad types do not set the weakly dominated price [MATH] , and consumers initially at the good types do not learn, pricing cannot get more competitive than with two firms. The outcome is the same as in Section |
More than two types (with higher quality implying lower cost) are conceptually similar to two, but notationally cumbersome. The worst type (highest cost, lowest quality) behaves like [MATH] . In particular, the worst types undercut each other in Bertrand fashion, until they price [MATH] above their cost. Consumers init... |
Two-dimensional types with combinations of cost and quality [MATH] [MATH] [MATH] and [MATH] are similar to the two-type case when cost and quality are negatively correlated. A type [MATH] cannot separate from [MATH] for any [MATH] in any equilibrium, because [MATH] can imitate any pricing strategy of [MATH] . The type ... |
If the correlation of cost and quality is positive, then the four-type model reduces to the case of two types with higher cost implying higher quality. Price signalling is then directed upward (the high-quality type sets a higher price). The race to the bottom does not occur. Each type sets a price weakly greater than ... |
If the correlation of cost and quality is zero, then signalling is impossible in either direction. Consumers expect the average quality after each price set in equilibrium and each type of firm sets its monopoly price given the expected quality. |
Suppose that the firms can advertise as well as signal by price. If ads reveal prices to some consumers, then competition increases and the good types cut prices below [MATH] . The bad types still set price [MATH] . If all consumers see both firms’ prices, then the good types Bertrand compete to price [MATH] |
If ads do not reveal prices, but are just wasteful signalling which for some reason is cheaper for the good type, then the results depend on the noisiness, timing and cost of the ads. If consumers cannot see the advertising expenditure, but must infer it from noisily observed ad quality and quantity, then ads seen befo... |
Suppose that ads are perfect signals of the money spent on them. Then the relative cost to the types per unit of ads vs per unit of price decrease determines which signalling channel the good type uses. If revealing the type via ads is relatively cheaper, then the good type sets its full-information monopoly price and ... |
If each firm trembles when setting price, and prices are the only way to signal, then the results depend on the trembles. Denote by [MATH] the probability that the consumers see price [MATH] when the firm tries to set [MATH] . A natural benchmark has [MATH] strictly decreasing in [MATH] , and [MATH] for all [MATH] Reas... |
3.1 Comparison to observable types In this section, the only difference from Section is that the type is not inferred from the price, but seen directly. The consumers initially at firm [MATH] see the price and type of firm [MATH] , but have to pay [MATH] to learn the price and type of firm [MATH] . In such a market, pr... |
Proposition 5 In any equilibrium with observable types, [MATH] , and if [MATH] , then [MATH] for [MATH] The idea for Proposition is that race to the top between the good types now continues at prices above [MATH] , as long as the profit increases in the price and consumers initially at a good type do not learn. If the ... |
The race to the top may end at the good type’s monopoly price or below that. If the race ends below [MATH] , then consumers initially at a good type learn and switch with positive probability. The bad type then gets positive demand, even when pricing above the other firm’s bad type. The captive customers of the bad typ... |
Conclusion The famous paradox of Diamond ( 1971 is that a market with multiple firms need not be competitive if consumers have to pay a cost to learn the prices of firms. However, as shown in the current paper, negatively correlated production cost and quality that are private information restore competitive pricing. T... |
The previous literature resolves the Diamond paradox assuming either (a) zero learning cost for a positive fraction of consumers, (b) that consumers observe multiple prices at once, (c) large private taste shocks, or (d) repeat purchases. The current paper models markets in which a given consumer purchases rarely, e.g.... |
If lower cost implies higher quality, then a low-cost firm would like to tell consumers about its cost level. A cheap talk message about low cost does not work, for the same reason as cheap talk about high quality has little effect. On the other hand, a low price is a credible signal, because it is differentially costl... |
Signalling by a low price resembles limit pricing, in which an incumbent tries to keep an entrant out of the market. The incumbent sets a low price to convince the entrant that the incumbent has a low cost and is likely to start a price war. The low price in limit pricing is anti-competitive. In the current work, the l... |
Appendix A Verification of the guessed equilibrium Consumers are clearly best responding to their beliefs in parts 3–5 of the guessed equilibrium. Beliefs in part 1 are consistent with part 2. It remains to check whether firms are best responding in part 2. First, downward deviations of type [MATH] are ruled out. The p... |
Lemma 6 In the guessed equilibrium, the profit of a type [MATH] firm from [MATH] is [EQUATION] Proof. The profit ( ) is derived from ( ) by substituting in the consumers’ strategies in the guessed equilibrium: [MATH] and [MATH] for consumers initially at [MATH] , because [MATH] and [MATH] Consumers with [MATH] buy from... |
If firm [MATH] is type [MATH] , then [MATH] With probability [MATH] , firm [MATH] is type [MATH] , in which case consumer [MATH] at firm [MATH] learns [MATH] with probability [MATH] and then buys if [MATH] |
Next, the technical Lemma simplifies ( ) by showing that if [MATH] and [MATH] for [MATH] , then [MATH] is a step function increasing in [MATH] |
Lemma 7 For customers initially at a type [MATH] firm, there exists [MATH] s.t. [MATH] for [MATH] and [MATH] for [MATH] Proof. Denote the type [MATH] firm by [MATH] . Due to [MATH] , in Definition (d), [MATH] may be dropped under the [MATH] w.l.o.g. If [MATH] , then the inequality is strict for all [MATH] |
If [MATH] , then [MATH] , so the first inequality in Definition (d) holds. If [MATH] , then [MATH] may be dropped under the [MATH] w.l.o.g. Then [MATH] and [MATH] imply that the first inequality in Definition (d) is strict for all [MATH] So if [MATH] , then for all [MATH] [MATH] Taking [MATH] ensures that [MATH] for [M... |
To prove [MATH] , note that [MATH] , so [MATH] . If [MATH] , then [MATH] . The [MATH] term in Definition (d) then ensures [MATH] |
Downward deviations by a type [MATH] firm are ruled out in the following Lemma. After that, the incentives of firm type [MATH] are discussed, and then the deviations of [MATH] to [MATH] are ruled out. |
Lemma 8 A type [MATH] firm’s best response to the strategies of other players in the guessed equilibrium satisfies [MATH] Proof. |
Based on Lemma [MATH] for all [MATH] , where [MATH] . Therefore ( ) reduces to [MATH] , with [MATH] independent of [MATH] The assumption [MATH] then implies [MATH] , because if [MATH] for all [MATH] , then for any [MATH] and [MATH] , we have [MATH] So type [MATH] optimally sets a price [MATH] |
Lemma 9 In the guessed equilibrium, a type [MATH] firm’s best response to the strategies of other players is [MATH] Proof. A type [MATH] firm clearly does not deviate to [MATH] , which is weakly dominated by [MATH] Consider [MATH] ’s deviations to [MATH] Parts 1 and 4 of the guessed equilibrium ensure that each custome... |
Having ruled out deviations by [MATH] , the final step (Lemma 10 ) is to eliminate upward deviations by a type [MATH] firm. Lemma 10 |
A type [MATH] firm’s best response to the strategies of other players in the guessed equilibrium satisfies [MATH] Proof. If a type [MATH] firm [MATH] sets [MATH] , then it gets zero demand in the guessed equilibrium, because [MATH] and the other firm [MATH] is expected to set price [MATH] . So [MATH] is not a profitabl... |
[MATH] , which holds by assumption. Combining Lemmas 10 the guessed equilibrium is verified. Appendix B Proofs omitted from the main text |
Proof of Lemma In any equilibrium, the incentive constraints (ICs) [MATH] and [MATH] hold for any [MATH] s.t. [MATH] . Demand and price are nonnegative and finite by definition. From [MATH] and [MATH] , we get [MATH] , so [MATH] |
If [MATH] and [MATH] , then [MATH] , so [MATH] Proof of Lemma Suppose that for both [MATH] there exists [MATH] s.t. [MATH] and [MATH] . Then [MATH] , otherwise [MATH] would deviate to put probability [MATH] on prices at which profit is positive. If [MATH] on or off the equilibrium path, then for all [MATH] [MATH] and b... |
Because [MATH] can imitate [MATH] at a strictly lower cost, we have [MATH] in any equilibrium, with strict inequality if [MATH] Next, the Intuitive Criterion is used to show that type [MATH] of firm [MATH] partially separates. Suppose that if [MATH] , then [MATH] (no separation of [MATH] ). Lemma implies that there is ... |
To apply the Intuitive Criterion, set [MATH] for all [MATH] . Check whether [MATH] wants to deviate to [MATH] By definition of [MATH] and [MATH] , there exists [MATH] (possibly equal to [MATH] ) s.t. [MATH] and [MATH] . Focus on the minimal such [MATH] . The definition of [MATH] implies [MATH] . Due to [MATH] , we have |
[MATH] Suppose [MATH] (implying [MATH] ). Then due to [MATH] , we have [MATH] and [MATH] , contradicting equilibrium. The remaining possibility is [MATH] (so [MATH] ). |
If [MATH] (which holds by assumption) and [MATH] , then a positive mass of consumers initially at [MATH] do not buy from [MATH] at [MATH] (consumer [MATH] strictly prefers not to). In that case, [MATH] reducing price to [MATH] and increasing belief to [MATH] strictly increases total demand [MATH] by at least [MATH] , w... |
If [MATH] and [MATH] , then [MATH] , i.e. [MATH] , contradicting equilibrium in the case [MATH] The Intuitive Criterion thus eliminates equilibria in which [MATH] , pooling among them. |
Because there exists [MATH] s.t. [MATH] and [MATH] , we have [MATH] with probability at least [MATH] . If either type of firm [MATH] sets price [MATH] , then [MATH] and [MATH] with the positive probability [MATH] , so firm [MATH] makes positive profit. |
Due to [MATH] , we can apply the Intuitive Criterion reasoning above to firm [MATH] to prove that type [MATH] of firm [MATH] also separates at least partially. |
Proof of Lemma If [MATH] , then by Lemma [MATH] for both [MATH] and [MATH] . The Intuitive Criterion then implies [MATH] for all [MATH] |
Suppose that [MATH] . If firm [MATH] raises price to [MATH] , then due to [MATH] , consumers initially at firm [MATH] still choose [MATH] . The customers at [MATH] who chose [MATH] anticipating [MATH] do not know about the deviation, so still choose [MATH] Upon learning [MATH] , a customer initially at [MATH] ’s type [... |
A type [MATH] customer initially at [MATH] chooses [MATH] only if [MATH] . From [MATH] , we get [MATH] and [MATH] . Due to [MATH] , if [MATH] , then [MATH] . The customers who might switch away from [MATH] after a price increase do not learn both prices, so have no choice of switching. |
On [MATH] , the profit of [MATH] is then given by ( ) and Lemmas prove that [MATH] for all [MATH] . ∎ Proof of Theorem By Lemma , there exist [MATH] s.t. [MATH] [MATH] [MATH] and [MATH] . Choose the maximal such [MATH] and assume [MATH] w.l.o.g. Clearly [MATH] . By Lemma , if [MATH] , then [MATH] , so if [MATH] , then ... |
By Definition (d),(e), consumer [MATH] initially at firm [MATH] charging [MATH] chooses [MATH] if [MATH] . W.l.o.g. [MATH] may be dropped under the [MATH] , due to [MATH] . Sufficient for [MATH] is then |
[MATH] equivalently [MATH] . This holds iff [MATH] , due to Bayes’ rule ( and [MATH] Consumers [MATH] always choose [MATH] . Due to [MATH] , all consumers initially at firm [MATH] choose [MATH] if [MATH] , i.e. [MATH] , which holds by assumption. If all consumers at type [MATH] of firm [MATH] choose [MATH] , then [MATH... |
If [MATH] , then deviating from [MATH] to the adjacent price [MATH] undercuts firm [MATH] ’s price [MATH] , increasing [MATH] by at least [MATH] . If [MATH] , then the deviation is profitable. Because [MATH] , we get [MATH] . Due to [MATH] , we can focus on either firm, so assume [MATH] w.l.o.g. This with [MATH] yields... |
If [MATH] , then [MATH] by Lemma . Finally, [MATH] implies [MATH] , so [MATH] Proof of Proposition Price [MATH] is available to type [MATH] , with [MATH] regardless of [MATH] , because [MATH] . Therefore [MATH] for [MATH] |
Denote [MATH] by [MATH] for [MATH] and assume w.l.o.g. [MATH] A customer type [MATH] initially at firm [MATH] who sees that the firm is type [MATH] and charges [MATH] chooses [MATH] if |
[MATH] . Type [MATH] sets [MATH] (which weakly dominates [MATH] ), firm [MATH] ’s type [MATH] sets [MATH] by assumption, and [MATH] , so |
[MATH] Sufficient for customer type [MATH] facing [MATH] not to learn is [MATH] , which holds if [MATH] So type [MATH] of firm [MATH] increases price to at least [MATH] Firm [MATH] was arbitrary, so the same reasoning applies to firm [MATH] |
# Source: arxiv 1806.00938 # Title: Program Synthesis from Visual Specification # Sections: all # Downloaded: 2026-03-03T02:29:50.298051+00:00 |
Program Synthesis from Visual Specification Abstract Program synthesis is the process of automatically translating a specification into computer code. Traditional synthesis settings require a formal, precise specification. Motivated by computer education applications where a student learns to code simple turtle-style d... |
Introduction The problem of program synthesis is an important one in AI. Synthesis has many settings, from the fully automated settings, where code is synthesized at the level of machine code, to the interactive, where synthesis assists a professional developer in an IDE. We focus on a novel synthesis setting where syn... |
Consider a student in an educational programming task, drawing an image with a turtle-style program. They may have an intention expressible as a trajectory they would like to draw, as shown in Figure |
(a). This trajectory can be considered a complete but noisy specification of the intended program. And they may have some program built up in their workspace, but this program may be “incomplete” or “buggy”. For example, |
Figure (b) shows a program the user might have composed with the intention of drawing Figure (a), along with the trajectory it actually creates. The user might be uncertain how to proceed in order to correct this. It is in this setting where we seek to formulate a synthesis task that can provide the solution shown in F... |
(c) where the arguments to both the repeat and turn blocks are corrected and a move block is added to yield the program nearest to the intended trajectory. |
In this paper, we describe how to synthesize code from such user-provided visual specification. We ultimately formulate the synthesis problem as an optimization problem suitable for combinatorial search. |
Setting 2.1 Programming Language We will consider the space [MATH] of turtle programs [EQUATION] Here Angle takes on values at increments of 30 degrees and Int takes on values from 2 to 5. Throughout the paper we will speak of elements of this space equivalently as either programs or blocks. As can be seen in Figure |
and elsewhere, blocks may be connected by being nested horizontally within repeat statements or vertically. If a block is connected vertically beneath another, we refer to it as a child block of its ancestor . If a block has no ancestors, we refer to it as a |
root block. We implement this turtle language using the Blockly visual block programming language and its editor. The semantics of this language are as follows: |
move statement translates the turtle in its current direction by some fixed magnitude. turn statement rotates the turtle by the specified number of degrees. |
repeat statement executes a subprogram some number of times. A user may position several such elements of [MATH] on their workspace as shown in Figure |
(a). Taking a more abstract perspective, we can define a workspace as a list of elements in [MATH] and we denote the space of workspaces by [MATH] . We call a set of points on the two-dimensional plane [MATH] |
trajectory . And we write [MATH] for the interpretation function which maps a workspace [MATH] to its trajectory [MATH] . This interpretation function can be thought of as executing each block in the workspace on the canvas in the order that it appears in workspace list. |
2.2 Editing Environment We can describe the user interface of an editor for our programming language through editing commands . These commands represent discrete mouse manipulations performable through Blockly. Note that for these commands to make sense we must label each block on the workspace with an identifying numb... |
The families of commands are: (1) adds a new block to the workspace, as a root block not connected to any other block on the workspace. The type parameter can be one of Move, Turn, or Repeat. |
(2) removes the block and all its child blocks. This command matches the Blockly semantics of dragging a block to the trash bin. |
(3) and (4) move a block and all its children to a new location. (3) moves a source block under a target block and connects them. If the target block has children, they are appended under the source block’s children. (4) is distinguished from (3) in that the target must be a repeat block, and it places the source block... |
(5) disconnects a block from its parent, making it into a root block on the workspace. (6) modifies the parameter values of blocks, such as the angle in the turn block or the integer in the repeat block. |
The user writes a program by applying a sequence of editing commands beginning from an empty workspace. That is, a command, when specialized by a choice of feasible values for its parameters, can be thought of as mapping a workspace [MATH] to a successor workspace [MATH] . For example, beginning at an empty workspace, ... |
(b): 1. Get a repeat block. 2. Get a move block. 3. Connect block 2 inside block 1. 4. Get a turn block. 5. Connect block 3 under block 2. |
6. Change 30 in block 3 to 120. This family of commands represents an abstraction of the editor’s capabilities, as the user would typically be manipulating the editing environment with keyboard and mouse. |
Problem Formulation Having described the programming language and editing environment, we are now in a position to formulate our synthesis problem as search. The user in our a programming environment intends to produce a trajectory [MATH] |
by means of a turtle program. Consider this as a search on a graph [MATH] whose vertices are the set of workspaces [MATH] . There is an edge from [MATH] to |
[MATH] in [MATH] if there is an editing command which produces workspace [MATH] when applied to workspace [MATH] . All edges have unit cost. We let [MATH] |
designate the weight of the shortest path from [MATH] to [MATH] in [MATH] . We will designate the initial state of their workspace by [MATH] |
We measure similarity of trajectories with Hausdorff distance. The Hausdorff distance is a commonly used metric for tasks in object matching and image analysis |
and serves as a natural metric for the quality of the fit of a candidate program to a trajectory. We denote the Hausdorff distance between sets of points [MATH] and [MATH] by |
[EQUATION] where [MATH] is the ordinary Euclidean distance. Synthesizing a good solution [MATH] from [MATH] for trajectory [MATH] |
involves a tradeoff between two types of distance or error. On the one hand, a candidate solution [MATH] has some distance from the target trajectory |
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