Method and system for matching entities in an auction

Abstract

A method for matching a first entity with at least one second entity selected from a plurality of second entities, comprising defining a plurality of multivalued scalar data representing inferential targeting parameters for the first entity and a plurality of multivalued scalar data of each of the plurality of second entities, representing respective characteristic parameters for each respective second entity; and performing an automated optimization with respect to an economic surplus of a respective match of the first entity with the at least one of the plurality of second entities, and an opportunity cost of the unavailability of the at least one of the plurality of second entities for matching with an alternate first entity.

Claims

1. A method of targeting a communication based on an expected performance, comprising: receiving a plurality of communications; determining a request for performance in conjunction with each respective communication; determining, for each respective communication, a set of possible targets for receiving the respective communication; selecting, for each respective communication, an optimum respective target from the set of possible targets, said selecting being optimized based on at least: an expected value of the determined performance of the request, a competitive alternate allocation of the respective possible target, a cost of allocation of the respective possible target to the respective communication, and an anticipated wait time for availability of the possible target; and producing at least one control signal for targeting the respective communication to the optimum respective target.

2. The method according to claim 1, wherein said selecting is further optimized based on a comprehensive evaluation of all competitive alternate allocations of the respective possible target.

3. The method according to claim 1, wherein the plurality of communications comprise voice communications.

4. The method according to claim 1, wherein the set of possible targets comprise call center agents.

5. The method according to claim 1, wherein the expected value comprises a revenue or profit metric.

6. The method according to claim 1, wherein the cost of allocation comprises a compensation rate of a respective target over an expected duration of the communication.

7. The method according to claim 1, wherein the performance of the request in conjunction with each respective communication is determined through interactive voice response.

8. The method according to claim 1, wherein said selecting comprises calculating a composite metric for each respective target, and selecting the target as the possible target having the best metric.

9. The method according to claim 1, wherein the control signal controls a telephone system.

10. The method according to claim 1, wherein the expected value of the determined performance of the request is based on at least a past history of a source of the respective communication.

11. A system for targeting a communication based on an expected performance, comprising: input ports configured to receive a plurality of communications; an automated communication process configured to determine a request for performance for each respective communication; at least one automated processor, configured to: determine a set of possible targets for each respective communication comprising a plurality of possible targets; select an optimal target based on at least a determination for each respective communication, and for each possible target, an expected value of the determined performance of the request, a competitive alternate allocation of the respective possible target, a cost of allocation of the respective possible target to the respective communication, and an anticipated wait time for availability of the possible target; and produce at least one control signal for controlling the respective communication with respect to allocation to the optimum respective target; and an output configured to produce a control signal for controlling communication routing of the communication to the optimum respective agent.

12. The system according to claim 11, wherein the at least one automated processor is further configured to select the optimum target further in dependence on an evaluation of a competitive alternate allocation of the respective possible target.

13. The system according to claim 11, wherein the plurality of communications comprise voice communications, and the input ports comprise voice communication ports.

14. The system according to claim 11, wherein the set of possible targets comprise call center agents, which are configured to handle the respective communication through a telephony routing system.

15. The system according to claim 11, wherein the expected value comprises a revenue or profit metric, and the cost of allocation comprises a compensation rate of a respective target over an expected duration of the communication.

16. The system according to claim 11, wherein the automated communication process comprises and interactive voice response system.

17. The system according to claim 11, further comprising a database of performance results, and wherein the expected value of the determined performance of the request is based on a record in the database associated with a source of the respective communication.

18. A method for pairing communication requests, comprising: receiving a series of requests for communication, and information identifying content or requestor-related characteristics associated with each respective requested communication; storing availability information and characteristics for a plurality of targets; generating a control signal for controlling at least establishment of the respective requested communication involving the requestor and a selected respective target, selectively optimized dependent on at least the information identifying content or requestor-related characteristics associated with the respective requested communication, a competitive alternate allocation of the selected respective target, an expected value of the respective requested communication, a cost of allocation of the respective communication to respective targets of the plurality of targets, and an anticipated wait time for availability of the respective targets of the plurality of targets; and outputting information selectively optimized dependent on the control signal.

19. The method according to claim 18, wherein the control signal is further generated in dependence on a comprehensive evaluation of competitive alternate allocations of respective targets of the plurality of targets.

20. The method according to claim 18, further comprising generating the control signal further in dependence on a history of the requestor.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) For a more complete understanding of the present invention and the advantages thereof, reference should be made to the following Detailed Description taken in connection with the accompanying drawings in which:

(2) FIG. 1 shows a first flow chart showing a skill routing method according to the present invention.

(3) FIG. 2 shows a second flow chart showing a skill routing method according to the present invention.

(4) FIG. 3 shows a first block diagram in accordance with the present invention.

(5) FIG. 4 shows a third flowchart in accordance with the present invention.

(6) FIG. 5 shows a second block diagram in accordance with the present invention.

(7) FIG. 6 shows a third block diagram in accordance with the present invention.

(8) FIG. 7 shows a forth flowchart in accordance with the present invention.

(9) FIG. 8 shows a fifth flowchart in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(10) The Detailed description of the invention is intended to describe relatively complete embodiments of the invention, through disclosure of details and reference to the drawings. The following detailed description sets forth numerous specific details to provide a thorough understanding of the invention. However, those of ordinary skill in the art will appreciate that the invention may be practiced without these specific details. In other instances, well-known methods, procedures, protocols, components, and circuits have not been completely described in detail so as not to obscure the invention. However, many such elements are described in the cited references which are incorporated herein by reference, or as are known in the art.

(11) For each agent, a profile is created based on manual inputs, such as language proficiency, formal education and training, position, and the like, as well as automatically, based on actual performance metrics and analysis, and used to create a skills inventory table. This process is generally performed in a high level system, such as a customer relations management system or human resources management system. A profile thus represents a synopsis of the skills and characteristics that an agent possesses, although it may not exist in a human readable or human comprehensible form.

(12) Preferably, the profile includes a number of vectors representing different attributes, which are preferably independent, but need not be. The profile relates to both the level of ability, i.e. expertise, in each skill vector, as well as the performance of the agent, which may be a distinct criterion, with respect to that skill. In other words, an agent may be quite knowledgeable with respect to a product line, but nevertheless relatively slow to service callers. The profile, or an adjunct database file, may also include a level of preference that call management has for the agent to handle transactions that require particular skills versus transactions that require other skills, or other extrinsic considerations.

(13) This table or set of tables is communicated to the communications server. Typically, the communications server does not create or modify the agent skills table, with the possible exception of updating parameters based on immediate performance. For example, parameters such as immediate past average call duration, spoken cadence, and other statistical parameters of a call-in-progress or immediately past concluded will be available to the communications server. These parameters, which may vary over the course of a single shift, may be used to adaptively tune the profile of the agent in real time. Typically, however, long term agent performance is managed at higher levels.

(14) FIG. 1 shows a flow chard of an incoming call routing algorithm according to a preferred embodiment of the present invention. A call is placed by a caller to a call center 301. The call is directed, through the public switched telephone network, although, calls or communications may also be received through other channels, such as the Internet, private branch exchange, intranet VOIP, etc. The source address of the call, for example the calling telephone number, IP address, or other identifier, is received to identify the caller 302. While the call is in the waiting queue, this identifier is then used to call up an associated database record 303, providing, for example, a prior history of interaction, a user record, or the like. The call waiting queue may be managed directly by the telephony server. In this case, since the caller is waiting, variable latencies due to communications with a separate call management system would generally not interfere with call processing, and therefore may be tolerated. In other instances, an interactive voice response (IVR) system may be employed to gather information from the caller during the wait period.

(15) In some instances, there will be no associated record, or in others, the identification may be ambiguous or incorrect. For example, a call from a PBX wherein an unambiguous caller extension is not provided outside the network, a call from a pay phone, or the like. Therefore, the identity of the caller is then confirmed using voice or promoted DTMF codes, which may include an account number, transaction identifier, or the like, based on the single or ambiguous records.

(16) During the identity confirmation process, the caller is also directed to provide certain details relating to the purpose of the call. For example, the maybe directed to press one for sales, two for service, three for technical support, four for returns, and five for other. Each selected choice, for example, could include a further menu, or an interactive voice response, or an option to record information.

(17) The call-related information is then coded as a call characteristic vector 304. This call characteristic is either generated within, or transmitted to, the communications server system.

(18) Each agent has a skill profile vector. This vector is developed based on various efficiency or productivity criteria. For example, in a sales position, productivity may be defined as sales volume or gross profits per call or per call minute, customer loyalty of past customers, or other appropriate metrics. In a service call, efficiency may be defined in terms of minutes per call, customer loyalty after the call, customer satisfaction during the call, successful resolution of the problem, or other metrics. These metrics may be absolute values, or normalized for the agent population, or both. The skill profile vector is stored in a table, and the profiles, which may be updated dynamically, of available or soon to be available agents, are accessed from the table (database) 305.

(19) Typically, the table 305 is provided or updated by a high level call center management system to the communications server system as the staffing assignments change, for example once or more per shift. Intra-shift management, such as scheduling breaks, may be performed at a low or high level.

(20) The optimization entails analysis of various information, which may include the caller characteristics, the call incident characterization, availability of agents, the agent profile(s), and/or various routing principles. According to the present invention, the necessary information is made directly available to the communications server, which performs an optimization to determine a best target, e.g., agent selection, for the caller.

(21) For example, if peak instantaneous efficiency is desired, for example when the call center is near capacity 306, more advanced optimizations may be bypassed and a traditional skill based call routing algorithm 307 implemented, which optimizes a short term cost-utility function of the call center 308. An agent who can optimally handle the call is then selected 309, and the call routed to that agent 310. The global (e.g., call center) factors may be accounted as a separate set of parameters.

(22) Thus, in order to immediately optimize the call routing, the general principle is to route the call such that the sum of the utility functions of the calls be maximized while the cost of handling those calls be minimized. Other types of optimizations may, of course, be applied.

(23) According to one optional aspect of the invention, the various routing principles discussed above explicitly value training as a utility of handling a call 311, and thus a long-term optimization is implemented 312. The utility of caller satisfaction is also weighted, and thus the agent selected is generally minimally capable of handling the call. Thus, while the caller may be somewhat burdened by assignment to a trainee agent, the call center utility is maximized over the long term, and call center agents will generally increase in skill rapidly.

(24) In order for the communications server system to be able to include these advanced factors, they must be expressed in a normalized format, such as a cost factor.

(25) As for the cost side of the optimization, the cost of running a call center generally is dependent on required shift staffing, since other costs are generally constant. Accordingly, a preferred type of training algorithm serves to minimize sub-locally optimal call routing during peak load periods, and thus would be expected to have no worse cost performance than traditional call centers. However, as the call center load is reduced, the call routing algorithm routes calls to trainee agents with respect to the call characteristics. This poses two costs. First, since the trainee is less skilled than a fully trained agent, the utility of the call will be reduced. Second, call center agent training generally requires a trainer be available to monitor and coach the trainee. While the trainer may be an active call center agent, and therefore part of the fixed overhead, there will be a marginal cost since the trainer agent might be assuming other responsibilities instead of training. For example, agents not consumed with inbound call handling may engage in outbound call campaigns.

(26) It is clearly apparent that the communications server system will have direct access to call center load data, both in terms of availability of agents and queue parameters.

(27) Thus, in a training scheme, an optimization is performed, using as at least one factor the value of training an agent with respect to that call 312, and an appropriate trainee agent selected 313.

(28) In order to provide proper training, the trainer and trainee must both be available, and the call routed to both 314. Generally, the trainee has primary responsibility for the call, and the trainer has no direct communication with the caller. Therefore, the trainer may join the call after commencement, or leave before closing. However, routing a call which requires two agents to be simultaneously available poses some difficulties. In general, the trainer is an agent capable of handling the entire call alone, while the trainee may not be. Therefore, the trainer is a more important participant, and the initial principle in routing the training call is to ensure that a trainer is available. The trainer may then await availability of an appropriate trainee, or if none is imminently available, handle the call himself or herself.

(29) On the other hand, where a specific training campaign is in place, and a high utility associated with agent training, then the availability of a specific trainee or class of trainees for a call having defined characteristics is particularly important. In that case, when an appropriate trainee is available, the call held in that agent's cue, and the call possibly commenced, awaiting a training agent's availability.

(30) If the training is highly structured, it is also possible to assign the trainer and trainee agents in pairs, so that the two are always available for calls together.

(31) The system according to the present invention may also provide reinforcement for various training. Thus, if a subset of agents receive classroom training on a topic, the server may target those agents with calls relating to that topic. For example, the topic may represent a parameter of a call characterization vector. In order to target certain agents for calls having particular characteristics, a negative cost may be applied, thus increasing the probability that the agent will be selected, as compared with an agent having a positive cost. By using a single cost function, rather than specific override, the system becomes resilient, since this allocation is not treated as an exception, and therefore other parameters may be simultaneously evaluated. For example, if a caller must communicate in a foreign language, and the agent does not speak that foreign language, then the system would not target the call to that agent, even if other factors weigh in favor of such targeting.

(32) The same techniques are available for outbound campaigns and/or mixed call centers. In this case, the cost of training is more pronounced, since agents idle for inbound tasks are generally assigned to outbound tasks, and thus the allocation of trainer agents and trainee agents generally results in both longer call duration and double the number of agents assigned per call. This cost may again be balanced by avoiding training during peak utility outbound calling hours and peak inbound calling hours; however, training opportunities should not be avoided absolutely.

(33) According to one embodiment of the invention, at the conclusion of a call, the caller is prompted through an IVR to immediately assess the interaction, allowing a subjective scoring of the interaction by the caller without delay. This information can then be used to update the stored profile parameters for both caller and agent, as well as to provide feedback to the agent and/or trainer. Under some circumstances, this may also allow immediate rectification of an unsatisfactory result.

(34) As shown in FIG. 3, a communications control system is shown comprising an input 502 receiving call classification information, the call classification information comprising a plurality of classification characteristics 501, a data structure 503 representing a plurality of distinct agent characteristics 504 for each of a plurality of agents; and a processor 509a, 509b, which may include a plurality of central processing units, for: (i) determining, with respect to the received call classification information, an optimum agent 515a, 515b, 515c, 515d, selected from sufficiently capable available agents for association with a call corresponding to call classification information, the determination of an optimum agent being based on a multifactorial optimization of (a) at least a non-binary weighted correspondence of said plurality of classification characteristics and (b) said plurality of distinct agent characteristics 504 for each of said plurality of agents 515a, 515b, 515c, 515d, and (ii) controlling a routing of a plurality of concurrent calls with a call router 512 in dependence on the determination. The processor may operate under control of a consolidated operating system 510. The determination and control by the processor may employ a common message queue 511 in the operating system 510. The process may maintain a data structure 503 representing skill weights 506 with respect to said call classification information 501, and applies said weights to determine an optimum agent selection. The processor may receive extrinsic perturbation information 508 independent of the plurality of classification characteristics and the plurality of distinct agent characteristics, to provide discrimination in control of call routing. A cost function may be provided for each agent 515a, 515b, 515c, 515d, the processor optimizing a cost-benefit outcome of a routing.

(35) A plurality of call classification vectors 501 may be received, the processor being adapted to determine, with respect to the received plurality of call classification vectors 501, an optimum association of the set of agents 15a, 515b, 15c, 515d, and calls having the associated call classification vectors 501. As shown in FIG. 4, a communications method is shown comprising: (a) receiving a plurality of concurrent communications, each having a plurality of associated classification factors 601; (b) storing information representing characteristics of a plurality of potential targets; (c) performing a multifactorial optimization to determine an optimum target for each of the plurality of concurrent communications based on the classification factors and the characteristics of the plurality of targets 602; and (d) routing at least one of the communications to a respective an optimum target, said performing step and said routing step being performed under control of a single computer operating system 605.

(36) The performing and routing may employ a common message queue in an operating system 607. A data structure representing skill weights with respect to the communication classification factors is applied to determine an optimum agent selection 604. The method may also include the step of perturbing the determining step to provide discrimination in routing 606. The determining step may comprise providing a cost function for each target, and optimizing a cost-benefit outcome of a routing 608.

(37) FIG. 4 shows a communications control system 701, comprising at least one programmable processor 702 executing instructions stored in a computer readable medium 703, the instructions being adapted to control the at least one programmable processor to execute: (a) a multithreaded operating system, providing support for applications and for passing messages between concurrently executing applications 703; (b) a communications control server application executing under said multithreaded operating system, for controlling concurrent real time communications 704; and (c) at least one dynamically linkable application, executing under said multithreaded operating system, communicating with said communications control server application to receive call characteristic data and perform a multifactorial optimization with respect to a plurality of target characteristics for each of a plurality of available communications targets, to resolve a an optimal communications target and transmit a resolved communications target, said communications control server application controlling a plurality of concurrent real time communications in dependence on the transmitted resolved communications target 706.

(38) As shown in FIG. 6, a communications matching system 801 is shown, comprising: (a) a plurality of communications channels for concurrently communicating with a plurality of entities 802; and (b) a communications router for defining a plurality of concurrent communications paths between sets comprising at least two of said entities, wherein said communications router conducts series of auctions to select respective communications paths from among a plurality of available competing paths, wherein said series of auctions determine winners based on a valuation function which is sensitive to both economic factors and non-economic factors, wherein the non-economic factors have an effect on auction outcome which changes over time, the non-economic factors being valued at the time of the auction 803.

(39) As shown in FIG. 7, a communications method is shown, comprising the steps of: (a) providing a plurality of communications channels for concurrently communicating with a plurality of entities 901; and (b) automatically concurrently routing communications between a plurality of communications channels based on the results of an automated auction which determines at least one winner based on a valuation function which is sensitive to both economic factors and non-economic factors 902, wherein the non-economic factors have an effect on auction outcome which changes over time, the non-economic factors being valued at the time of the auction.

(40) The communications channels may be of a first type and a second type, the communications being routed between a user of at least one of a first type of communications channel and a user of at least one of a second type of communications channel 903. The non-economic factors may comprise an optimality of matching a profile representing a user of a communications channel of the first type with a profile of a user of a communications channel of a second type 904.

(41) The economic factors may compensate for a suboptimiality of a matching of profiles to perturb a non-economic optimal matching from the auction 905.

(42) As shown in FIG. 8, a communications control method is shown, comprising the steps of: (a) receiving call classification information for each of a plurality of calls, the call classification information comprising a plurality of classification characteristics 1001; (b) representing a plurality of agent characteristics for each of a plurality of agents 1002; (c) determining, with respect to the received call classification information associated with a plurality of concurrent calls, an optimum set of concurrent mutually exclusive associations of the set of agents with the plurality of calls, the determining being dependent on at least a multifactorial optimization of (i) the weighted correspondence of the plurality of classification characteristics for each of the respective plurality of calls, and (ii) the plurality of agent characteristics for each of said plurality of agents 1003; and (d) controlling a concurrent call routing of the plurality of calls in dependence on the determination 1004.

Example 1

(43) Each agent is classified with respect to 10 skills, and each skill can have a weight of 0 to 127. The skill weights may be entered manually by a supervisor, developed adaptively, or provided by other means. These are sent as a parameter file to the communications server.

(44) A rule vector specifies a normalized contribution of each skill to apply to the total. This rule vector, for example, represents the call characteristic vector. Thus, attributes of the call and the status of the system are analyzed to generate this rule vector. There can be more than one rule vector defined in a project (split), or a rule can be setup in a per call basis. Generally, routing with predefined rules is much more efficient than routing with rules in a per call bases. When a call needs to be routed to an agent, the rule vector is applied to the skills of the available agents and a score is derived for each agent. The agent with the highest score is assigned the call, as shown in Table 1.

(45) As shown in Table 1, Agent 1 would be selected, since this is the highest score.

(46) In this example, it is presumed that all selections have the same cost, and therefore the utility only varies. Thus, the agent with the highest utility function is the optimal selection.

Example 2

(47) The conditions below are the same as in Example 1, except two new factors are provided, Ac1 and Ac2. The Preliminary Score is calculated as the sum of the products of the Rule Vector and the Agent Vector. The Final Score is calculated as (Ac1sum)+Ac2.

(48) In this case, Ac1 represents an agent-skill weighting cost function, while Ac2 represents an agent cost function. Since we select the maximum value, more expensive agents have correspondingly lower cost values.

(49) As can be seen in Table 2, Agent 5 is now optimum.

Example 3

(50) In this example, a limiting criterion is imposed, that is, only agents with a skill score within a bound are eligible for selection. While this may be implemented in a number of ways, possibly the simplest is to define the range, which will typically be a lower skill limit only, below which an agent is excluded from selection, as a preliminary test for availability.

(51) As noted below in Table 3, the screening criteria may be lower, upper or range limits. In this case, the screening process excludes agents 2, 3, and 5, leaving agents 1 and 4 available. Of these two choices, agent 1 has the higher score and would be targeted. (Note: 2, 3, 5 excluded, 1, 4 available).

Example 4

(52) In this example, the optimization seeks to optimize the placement of 5 incoming calls to 5 agents. As shown in Table 4, each caller is represented by a different call vector, and each agent by a distinct skill vector. The optimization therefore seeks the maximum utility from the respective possible pairings.

(53) TABLE-US-00001 TABLE 1 Agent Agent Agent Agent Agent Rule vector 1 2 3 4 5 20% Skill 1 20 5 3 5 4 5% Skill 2 3 3 3 3 3 10% Skill 3 10 6 9 10 10 15% Skill 4 43 50 33 46 25 3% Skill 5 7 2 9 2 8 7% Skill 6 5 8 5 8 9 20% Skill 7 2 3 4 2 2 8% Skill 8 64 80 29 45 77 5% Skill 9 4 5 4 1 2 7% Skill 10 9 3 8 3 6 100% Score 18.51 17.33 11.1 13.93 13.65

(54) TABLE-US-00002 TABLE 2 Rule Vector Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Ac1 0.4 0.55 0.45 0.7 0.6 Ac2 6 3 6.8 2 5.5 20% Skill 1 20 5 3 5 4 5% Skill 2 3 3 3 3 3 10% Skill 3 10 6 9 10 10 15% Skill 4 43 50 33 46 25 3% Skill 5 7 2 9 2 8 7% Skill 6 5 8 5 8 9 20% Skill 7 2 3 4 2 2 8% Skill 8 64 80 29 45 77 5% Skill 9 4 5 4 1 2 7% Skill 10 9 3 8 3 6 100% Prelim Score 18.51 17.33 11.1 13.93 13.65 Final Score 13.40 12.53 11.80 11.75 13.69

(55) TABLE-US-00003 TABLE 3 Rule Vector Exclude Agent Agent Agent Agent Agent Min Skill Max Skill Agent 1 2 3 4 5 Ac1 0.4 0.55 0.45 0.7 0.6 Ac2 6 3 6.8 2 5.5 20% 0% 25% Skill 1 20 5 3 5 4 5% Skill 2 3 3 3 3 3 10% Skill 3 10 6 9 10 10 15% 40% 100% 3, 5 Skill 4 43 50 33 46 25 3% Skill 5 7 2 9 2 8 7% Skill 6 5 8 5 8 9 20% Skill 7 2 3 4 2 2 8% 30% 75% 2, 3, 5 Skill 8 64 80 29 45 77 5% Skill 9 4 5 4 1 2 7% Skill 10 9 3 8 3 6 100% Prelim Score 18.51 17.33 11.1 13.93 13.65 Final Score 13.40 12.53 11.80 11.75 13.69

(56) TABLE-US-00004 TABLE 4 Rule Vector Rule Vector Rule Vector Rule Vector Rule Vector Agent Agent Agent Agent Agent SKILL 1 2 3 4 5 1 2 3 4 5 1 20% 25% 17% 20% 14% 20 5 3 5 4 2 5% 10% 5% 5% 3% 3 3 3 3 3 3 10% 15% 20% 10% 8% 10 6 9 10 10 4 15% 10% 5% 5% 5% 43 50 33 46 25 5 3% 0% 5% 8% 1% 7 2 9 2 8 6 7% 10% 13% 10% 7% 5 8 5 8 9 7 20% 10% 5% 10% 20% 2 3 4 2 2 8 8% 4% 8% 4% 8% 64 80 29 45 77 9 5% 8% 13% 18% 23% 4 5 4 1 2 10 7% 8% 9% 10% 11% 9 3 8 3 6 100% 100% 100% 100% 100% Rule 1 18.51 17.33 11.1 13.93 13.65 Rule 2 15.4 12.39 8.72 10.77 10.12 Rule 3 15.25 13.31 8.97 10.54 12.71 Rule 4 12.74 9.91 7.6 7.89 8.98 Rule 5 13.69 12.83 8.24 9.03 11.09

(57) TABLE-US-00005 TABLE 6 Rule Rule Rule Rule Rule Agent Agent Agent Agent Agent SKILL Vector 1 Vector 2 Vector 3 Vector 4 Vector 5 1 2 3 4 5 Caller time factor 3 3.5 2.75 4 10 Agent 0.59 0.68 1 0.86 0.79 Cost Agent 1.3 1.3 1 1.1 1.2 time factor 1 20% 25% 17% 20% 14% 20 5 3 5 4 2 5% 10% 5% 5% 3% 3 3 3 3 3 3 10% 15% 20% 10% 8% 10 6 9 10 10 4 15% 10% 5% 5% 5% 43 50 33 46 25 5 3% 0% 5% 8% 1% 7 2 9 2 8 6 7% 10% 13% 10% 7% 5 8 5 8 9 7 20% 10% 5% 10% 20% 2 3 4 2 2 8 8% 4% 8% 4% 8% 64 80 29 45 77 9 5% 8% 13% 18% 23% 4 5 4 1 2 10 7% 8% 9% 10% 11% 9 3 8 3 6 100% 100% 100% 100% 100% Rule 1 72.189 58.80069 19.647 31.71861 36.855 Rule 2 70.07 49.045815 18.0068 28.610505 31.878 Rule 3 54.51875 41.3974275 14.553825 21.999615 31.45725 Rule 4 66.248 44.83284 17.936 23.95404 32.328 Rule 5 177.97 145.1073 48.616 68.5377 99.81

(58) TABLE-US-00006 TABLE 5 Combinatorial analysis of agents vs. callers 58.85 59.52 58.77 59.58 60.68 58.79 58.04 58.85 60.28 57.72 58.12 58.93 60.42 57.86 59.01 60.15 59.26 56.7 59.96 58.18 57.88 58.55 58.88 60.66 59.71 57.82 58.15 59.93 59.31 56.75 55.41 57.19 60.53 57.97 56.3 57.33 60.34 57.78 57.25 55.36 60.13 61.28 59.25 60.06 61.96 61.33 59.3 60.11 62.04 60.26 58.9 59.71 60.9 59.12 59.79 60.93 59.74 57.96 58.63 58.96 60.24 49.54 56.54 58.32 62.07 58.99 56.96 58.74 59.33 57.92 56.56 58.34 58.19 56.78 57.45 58.48 58 56.59 57.26 56.51 58.66 59.81 60.14 62.42 60.49 59.86 60.19 62.47 60.57 58.79 57.45 59.73 61.79 60.01 58.34 58.59 62.1 60.32 58.65 56.62 59.74 58.07 57.32 59.6 61.57 58.49 57.74 60.02 58.83 57.42 57.82 60.1 58.97 57.56 58.71 58.96 59.28 57.87 59.02 56.99

(59) TABLE-US-00007 TABLE 7 Combinatorial Analysis 259.5527 255.3573 256.8383 289.6307 267.1886 254.8785 256.3595 289.1519 255.0903 246.9756 236.6543 232.4589 235.0236 290.7144 244.2902 231.9801 234.5448 290.2356 232.1919 224.0773 260.9804 259.9429 259.9962 292.7886 268.6163 260.932 260.9853 293.7777 259.6759 253.0292 239.1658 163.4004 231.9914 287.6822 246.8016 234.7961 234.8494 290.5402 231.6711 226.8933 224.1784 223.1409 225.7056 295.3001 231.8143 224.13 226.6947 296.2892 222.8739 216.2272 225.2621 218.0345 219.5155 289.1099 232.898 220.8925 222.3735 291.9679 217.7675 212.9896 247.4191 280.2116 264.8651 256.7504 255.7129 291.0701 309.1051 300.9904 367.4349 302.5176 219.4143 275.1051 243.0504 234.9358 227.7081 284.8799 310.1888 302.0741 339.4301 296.3275 248.887 281.6795 268.023 261.3763 257.1808 292.538 312.263 305.6163 301.4208 303.9855 222.7511 278.4419 240.0182 235.2403 231.0449 288.2167 307.1566 302.3787 298.1833 299.6643 211.5642 281.1587 233.7324 227.0857 219.858 289.5057 314.7744 308.1277 300.9 300.9533 213.4331 283.0276 227.5423 222.7644 221.7269 291.3746 308.5843 303.8064 302.7689 302.8222

(60) Using a combinatorial analysis, as shown in Table 5, the maximum value is 62.42, which represents the selection of agent 1/caller 1; agent 2/caller 5; agent 3/caller 4; agent 4, caller 2; and agent 5, caller 3.

Example 5

(61) Similar to Example 4, it is also possible to include an agent cost analysis, to provide an optimum cost-utility function. As in Example 2, the cost factors are reciprocal, since we select the largest value as the optimum. Likewise, time factors are also reciprocal, since we seek to minimize the time spent per call. In this case, shown in Table 6, the cost analysis employs three additional parameters: the agent cost, a value representing the cost of the agent per unit time; a value representing an anticipated duration of the call based on the characteristics of the caller; and a value representing the anticipated duration of the call based on characteristics of the agent.

(62) As can be seen in Table 7, the maximum value is 314.78, which corresponds to a selection of: Agent 1/Call 5; Agent 2/Call 1; Agent 3/Call 4; Agent 4/Call 2; and Agent 5/Call 3.

(63) Therefore, it is seen that the optimum agent/caller selection is sensitive to these cost factors.

(64) It is also seen that, while the analysis can become quite complex, the formulae may be limited to evaluation of simple arithmetic functions, principally addition and multiplication, with few divisions required. Thus, these calculations may be executed efficiently in a general purpose computing environment.

Example 6

(65) Geographic information may be used as a basis for communications routing. Mobile phones are or will be capable of geolocation, meaning that the location of the handset may be automatically determined in real time and communicated. Likewise, a location of landlines can typically be determined. There are a number of instances where this information may then advantageously be used to route calls. For example, a call to a national pizza delivery chain toll free number or central facility may be automatically routed to a geographically proximate local franchisee, or, if a number are available, to one of a qualified group. It is noted that while the communications are preferably voice communications, other type of communications may be supported.

(66) However, it is also possible to perform evaluation of more complex algorithms in order to determine a set of communications partners. For example, a geographic factor, a past history, and/or user profile may be available to describe the caller. This information may provide, for example, a preferred language, a contact report (identifying likely issues), demographic information, and user personality (as determined from a prior communication). Likewise, an interactive voice or keypad response system can glean further information to determine the issues involved in the call. Using this information, a vector may be provided describing the caller and the likely issues of the call, which may then be used to optimize a targeting of the call to available recipients. The maintenance of vectors to describe available call targets is described above.

(67) In cases where multiple recipients are available and have, within a reasonable range, equivalent or super-threshold qualifications or suitability to receive the call, it may be appropriate for the potential recipients to compete for the call. That is, the optimization of targeting (e.g., pairing of a caller and callee) includes an economic component, optionally with a non-economic component. For example, the potential recipients each submit a bid for the call, with the call being routed to the auction winner (which may be a payment to or from the recipient, depending on the circumstances of the auction) at, for example, a first or second price, according to the auction rules.

(68) In a typical case, the routing server has a direct and prearranged financial arrangement with the bidders, and the auction process does not directly involve the caller. On the other hand, other cases allow the caller to be involved in the auction as a buyer or seller, with the communications router serving only in the capacity of auctioneer, and not a principal to the auction.

(69) In cases where the potential recipients do not all have equivalent qualifications, a normalization function may be applied to correct the bids. For example, a potential recipient with a 60% match with the required qualification profile might have to bid 50% more than a potential recipient with a 90% match, assuming that the matching function linearly corresponds with an economic factor; otherwise, a non-linear normalization may be applied. This is equivalent to providing that the value applied to determine the auction winner includes a component representing an economic value and a component representing a non-economic value, e.g., a match or optimality score for the call, which is determined for each bidder to determine the winner. The bidder in this case may either have knowledge of the match score, or may bid blind.

(70) In a commission based system, for example, an agent with a higher sales average performance might have to bid a lower amount than an agent with lower performance, the difference being an amount which tends to equalize (but not necessarily completely equalize) the anticipated payoff from the call, thus incentivizing higher sales performance. In any case, the communications router (or a separate system which communicates with the communications router in some embodiments) evaluates the bids including both economic and non-economic components, determines the winning bidder, and determines the communications path(s).

(71) In another embodiment, a group of agents within a call center have performance goals for a shift, with possible gradation between agents of the goals based on compensation, seniority, etc. The agents are within a queue, in which the default is a sequential selection of available agents. However, an agent may seek to take a break, and therefore bids for a lower position within the queue. Likewise, an agent may find him or herself behind in performance, and wish to bid for higher placement within the queue. As discussed above, the bid cost or perturbation effect may be normalized based on a variety of factors and schemes, including the optimality of matching. In this scheme, the auction may be economic or non-economic. In a non-economic scheme, each agent is provided with a set of bid units, for example 100 per shift. The bid units may then be applied to advance within the queue, or even traded with another agent (although this possibility leaves open the issue of undesired indirect real economic effects, since the trade may involve extrinsic value).

(72) Another possibility is the ad hoc formation of chat groups. In this case, the composition of the group is optimized based on the respective profile vectors of the members. In some cases, the ideal or optimum is minimum variance of the vectors, but in other cases optimality may require complementary components. Assuming multiple chat groups and multiple callers, there may be a market economy for matching a caller with a group. In such a scenario, a VCG type auction may be conducted, with the composition of each group allocated based on an optimization of bid values. An example of this is a sports chat line. A number of fans and sports celebrities contact a call center and are identified and a profile applied. Using market principles, the groups are formed to maximize the utility aggregate functions. Thus, a group of high rollers may gain the benefit of a superstar, while neophytes may only communicate with a rookie, with the set of groups optimized to achieve maximum utility.

(73) An automated chat system may also be used for dating services, adult theme entertainment, business services, consumer services, or the like. In these systems, the communications router typically taxes some of the economic surplus generated by the system, in a real economic form, while benefiting the various classes of user.

(74) It is noted that the auction may involve transfer of real economic benefits, or a synthetic economy constructed within a closed system. For example, micropayment technologies may be employed to authorize and convey the value between entities, even through an open network, without having to trust all entities within the chain of custody.

(75) The bidding may be a volitional real time event, allowing those involved to make decisions on the spot; but more typically, a bidder will define a personal value function, which is then used in an automated auction process. The bidder will therefore provide an indirect control over the bidding on his or her behalf, for example using feedback to tune the attributed value function to a desired value. In auction types where broadcast of a true value is a dominant strategy, the function itself may be presented as a bid (assuming that the auctioneer has sufficient information to evaluate the function), otherwise, it may be evaluated under the circumstances and a normalized value transmitted. The auctioneer is, in this case, the communications arbitrator or switch. In a successive price auction, the value function itself is preserved, although the dropout pattern may be noted, allowing an estimation of the value function of competitors.

(76) It should be clear that there are many possible scenarios which allow callers and/or potential recipients to compete for a connection, and therefore a large variety of auction types may be implemented accordingly.

(77) The present system differs from a known telecommunications auction in that, for example, it is sensitive to user characteristics, and does not treat each communications line as a simple commodity.

(78) From the above description and drawings, it will be understood by those of ordinary skill in the art that the particular embodiments shown and described are for purposes of illustration only and are not intended to limit the scope of the invention. Those of ordinary skill in the art will recognize that the invention may be embodied in other specific forms without departing from its spirit or essential characteristics. References to details of particular embodiments are not intended to limit the scope of the claims.

(79) It should be appreciated by those skilled in the art that the specific embodiments disclosed above may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims.

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(89) Extensive Form Games without Perfect Information Kreps and Wilson, Sequential Equilibrium. Econometrica, 1983. Van Damme, Stable Equilibria and forward induction. JET 1989.

(90) Strategic Information Transmission Crawford and Sobel, Strategic information transmission. Econometrica 1982. Cho and Kreps, Signalling games and stable equilibria. QJE 1987 Mailath, Okuno-Fujiwara and Postlewaite, Beleif based refinements in signalling games. JET 1993 Milgrom and Roberts, Limit pricing and entry under incomplete information: an equilibrium analysis. EMA 1982 (pages 443-459) Cho and Sobel, Strategic Stability and uniqueness in signalling games. JET 1990. Farrell, Meaning and credibility in cheap talk games. GEB Milgrom and Roberts, Limit pricing and entry under incomplete information, an equilibrium analysis, EMA 1982 Milgrom, Good news and bad news, representation and applications, Rand.

(91) Folk Theorems for Repeated Games Dilip Abreu. On the theory of infinitely repeated games with Discounting. Econometrica 1988 Benoit and Krishna. Finitely Repeated games. Econometrica, 1985. James Friedman. A noncooperative equilibrium for supergames. REStud 1971. James Friedman. Cooperative equilibria in finite horizon supergames. JET 1985. Fudenberg and Maskins. The Folk Theorem in repeated gmaes with discounting or with incomplete information. Econometrica 1986. Roy Radner. Collusive Behavior in non-cooperative epsilon equilibria in oligopolies with long but finite lives. JET 1980. Ariel Rubinstein. Equilibrium in supergames with the overtaking criterion. JET 1977.

(92) Renegotiation Benoit and Krisna, Renegotiation in finitely repeated games. EMA 1993 Bergin and MacCleod, Efficiency adn renegotiation in repeated games. JET 1993 Andreas Blume, Interplay communication in repeated games. GEB 1994 Geir Asheim, Extending renegotiation proofness to infinite horizon games. GEB 1991 Bernheim and Ray, Collective dynamic consistency in repeated games. GEB 1989 Farrel and Maskin, Renegotiation in Repeated Games. GEB 1989

(93) Cooperative Game Theory Freidman, Game theory with applications to economics chapter 6 and 7 Nash, The Bargaining problem. EMA 1950 Kalai and Smordinski, Other Solutions to Nash's problem. EMA 1975

(94) Noncooperative Bargaining Rubinstein, Perfect equiibrium in a bargaining model. EMA 1982 Joel Watson, Alternating offer bargaining with two sided incomplete information. REStud 1999

(95) Reputation Kreps, Milgrom, Roberts and Wilson, Reputation and imperfect information: predation, reputation and entry deterence: rational cooperation in the finitely repeated prisoner's dilemna. JET 1981 Aumann and Sorin, Cooperation and Bounded recal. GEB 1989 Klaus Schmidt, Reputation and equilibrium characterization in repeated games with conflicting interests, Econometrica 1993, 325-352 Cripps and Thomas, Reputation and Commitment in Two person games without discounting, EMA, 1995, 1401-1420 Joel Watson, A reputation refinement withough equilibrium, EMA 1993, 199-206 Celentani, Fudenberg and Levine, Maintaining a reputation against a long lived opponent EMA 1996, 691-704

(96) Evolutionary Game Theory Vince Crawford, An Evolutionary interpretation of VHBB's experimental results on coordination. GEB 1991 Gilboa and Matsui, Social Stability and Equilibrium, EMA 1991 Kandori, Mailath and Rob, Learning, Mutation, and Long Run Equilibria in games, EMA 1993. Peyton Young, An Evolutionary Model of Bargaining, JET 1993 Peyton Young, The Evolution of Conventions, EMA 1993 Larry Samuelson, Stochastic Stability with alternative best replies. JET Noldeke and Samuelson, The Evolution of Backwards and Forwards Induction, GEB 1993 Jack Robles, An Evolutionary Folk Theorem For Finitely Repeated Games CU WP 99 Kim and Sobel, An Evolutionary Approach to Preplay Communication EMA 1995

(97) General Game Theory Bierman H. S. & Fernandez L., Game Theory with Economic Applications, Addison-Wesley, 1993. Dixit A., & Nalebuff B., Thinking Strategically: the Competitive Edge in Business, Politics, and Everyday Life, New York: Norton, 1991. McMillan J., Games, Strategies, and Managers, Oxford: OUP, 1992. Baird D. G., Gertner R. H., and Picker R. C., Game Theory and the Law, Harvard U. P., 1994. Rasmusen E., Games and Information: An Introduction to Game Theory, Oxford: B. Blackwell, 2nd edition, 1994. Ghemawat P., Games Businesses Play: Cases and Models, New York: Wiley, 1995. Gardner R., Games for Business and Economics, New York: Wiley, 1995.

(98) Strategic Decision Making Dixit & Nalebuff, Intro; Ch2 Anticipating your rival's response; Ch3 Seeing through your rival's response. Barnett, F. W. Making game theory work in practice, Wall Street Journal, 1995. Bierman & Fernandez, Ch5 Nash equilibrium I, Ch11 Nash equilibrium II O'Neill B., International escalation and the dollar auction, Journal of Conflict Resolution, 1986. Schelling T. C., Ch7 Hockey helmets, daylight saving, and other binary choices, in his Micromotives and Macrobehavior, NY: Norton, 1978. Marks R. E., Competition and common property, 1998. McMillan J., Ch3 Understanding cooperation and conflict. McAfee R. P. & J. McMillan, Competition and game theory, Journal of Marketing Research, 1996. Baird, Gertner, & Picker, Ch1 Simultaneous decision-making and the normal form game. Gardner, Ch1 Introduction, Ch2 Two-person games, Ch16 Voting games. Rasmusen, Ch1 The rules of the game. Schelling T. C., What is game theory? in his Choice and Consequence: Perspectives of an Errant Economist, Camb.: Harvard UP, 1980.

(99) Decision AnalysisGames Against Nature Apocalpse maybe, and An insurer's worst nightmare, The Economist, 1995/96 Bierman & Fernandez, Chs 1-3. Ulvila J. W. & R. Brown, Decision analysis comes of age, Harvard Buisness Review 1982. Howard R. A., Decision analysis: practice and promise, Management Science, 1988. Clemen R. T., Making Hard Decisions: An Introduction to Decision Analysis, Belmont, Calif.: Duxbury, 1996. Samson D., Chs 2-6, 11, Managerial Decision Analysis, Chicago: R. D. Irwin, 1988.

(100) Strategic Moves Dixit & Nalebuff, Ch5 Strategic moves. Brams S. J. & J. M. Togman, Cooperation through threats: the Northern Ireland case, PS: Political Science & Politics, March 1998. Gardner, Ch4 n-person games, Ch5 Non-cooperative games. Colman A. M., Ch8 Multi-person games: social dilemmas, in his Game Theory and Experimental Games, Oxford: Pergamon, 1982. Kay J., Ch3 Co-operation and Co-ordination, in his Foundations of Corporate Success: How Business Strategies Add Value, Oxford: OUP, 1993. Brams S. J., Ch1 International relations games, in Game Theory and Politics, NY: Macmillan, 1975.

(101) Credible Commitment Dixit & Nalebuff, Ch6 Credible commitments. Bierman & Fernandez, Ch23 Subgame-perfect equilibrium Rasmusen, Ch4.1 Subgame perfection. Gardner, Ch6 Credibility and subgame perfection. Ghemawat, Ch3 Preemptive capacity expansion in the titanium dioxide industry.

(102) Repetition and Reputation Dixit & Nalebuff, Ch4 Resolving the Prisoner's Dilemma; Ch9 Cooperation and coordination. Nowak, M., R. May, & K. Sigmund, The arithmetic of mutual help, Scientific American, 1995 Hofstadter D., Ch29 The Prisoner's Dilemma computer tournaments and the evolution of cooperation, in his Metamagical Themas, Penguin, 1985. Marks R. E., Midgley F D. F., & Cooper L. G., Adaptive behaviour in an oligopoly, in Evolutionary Algorithms in Management Applications, ed. by J. Biethahn & V. Nissen, (Berlin: Springer-Verlag), 1995. Baird Gertner & Picker, Ch2 Dynamic interaction and the extensive-form game, Ch5 Reputation and repeated games. Gardner, Ch7 Repeated games, Ch8 Evolutionary stability and bounded rationality. Rasmusen, Ch4 Dynamic games and symmetric information, Ch5 Reputation and repeated games with symmetric information.

(103) Unpredictability Dixit & Nalebuff, Ch7 Unpredictability; Ch8 Brinkmanship. Bierman & Fernandez, Ch11.9 Gardner, Ch3 Mixed strategies. Rasmusen, Ch3 Mixed and continuous strategies.

(104) Bargaining Dixit & Nalebuff, Ch10 The voting strategy; Ch11 Bargaining. McMillan, Ch5 Gaining bargaining power; Ch6 Using information strategically. Elster J., Ch14 Bargaining, in Nuts and Bolts for the Social Sciences, Camb.: CUP, 1989 Murnighan J. K., Game's End, Chapter 15 in his: Bargaining Games: A New Approach to Strategic Thinking in Negotiations, NY: William Morrow, 1992. Bierman & Fernandez, Ch6 Bargaining. Schelling T. C., Ch2 Essay on bargaining, in The Strategy of Conflict, Camb.: Harvard UP, 1980. Baird Gertner & Picker, Ch7 Noncooperative bargaining Gardner, Ch12 Two-person bargains. Ch14 n-person bargaining and the core. Rasmusen, Ch11 Bargaining. Brams S. J., Negotiation Games: Applying Game Theory to Bargaining and Arbitration, NY: Routledge, 1990.

(105) Using Information Strategically McMillan, Ch6 Using information strategically Bierman & Fernandez, Ch17 Bayesian equilibrium, Ch19 Adverse selection and credit rationing Rasmusen, Ch2 Information P-13 Baird Gertner & Picker, Ch4 Signalling, screening, and nonverifiable information Gardner, Ch9 Signaling games.

(106) Bidding in Competition Revenge of the nerds, It's only a game, and Learning to play the game, The Economist, 1994 Landsburg S. E., Cursed winners and glum losers, Ch18 of his The Armchair Economist: Economics and Everyday Life, New York: The Free Press, 1993. Norton, R., Winning the game of business, Fortune, 1995, Koselka, R., Playing poker with Craig McCaw, Forbes, 1995, Dixit & Nalebuff, Ch12 Incentives. McMillan, Ch11 Bidding in competition McAfee R. P. & J. McMillan, Analyzing the airwaves auction, Journal of Economic Perspectives, 1996 R. Marks, Closed tender vs. open bidding auctions, 22 Dec. 1994. The Economist, Secrets and the prize, 12 Oct. 1996, p. 98. Scientific American, Making honesty pay, January 1997, p. 13. Gardner, Ch11 Auctions. Brams S. J. & A. D. Taylor, Fair division by auctions, Ch9 of their Fair Division: From Cake-Cutting to Dispute Resolution, Cambridge: CUP, 1996. Rasmusen, Ch12 Auctions.

(107) Contracting, or the Rules of the Game Kay, Ch4 Relationships and contracts. Dixit & Nalebuff, Ch12 Incentives. McMillan, Ch8 Creating incentives; Ch9 Designing contracts; Ch10 Setting executives' salaries. Williamson O. E., Strategizing, economizing, and economic organization, Strategic Management Journal, 1991. Bierman & Fernandez, Ch7 Involuntary unemployment. Gardner, Ch10 Games between a principal and an agent. Milgrom P. & Roberts J., Ch5 Bounded rationality and private information; Ch6 Moral hazard and performance incentives. Economics, Organization and Management, Englewood Cliffs: Prentice-Hall, 1992.

(108) Choosing the Right Game: Co-Opetition Brandenburger A. M. & B. J. Nalebuff, The right game: using Game Theory to shape strategy, Harvard Business Review, 1995 mayet.som.yale.edu/coopetition/index2.html Koselka R., Businessman's dilemma, and Evolutionary economics: nice guys don't finish last, Forbes, Oct. 11, 1993. Brandenburger A. M. & B. J. Nalebuff, Co-opetition: 1. A revolutionary mindset that combines competition and cooperation; 2. The Game Theory Strategy that's changing the game of business. New York: Currency Doubleday, 1996. Brandenburger A. M. & Harborne W. S. Jr., Value-based business strategy, Journal of Economics and Management Strategy, 5(1), 1996. Baird Gertner & Picker, Ch6 Collective action, embedded games, and the limits of simple models. Morrow J. D., Game Theory for Political Scientists, Princeton: P.U.P., 1994. Casson M., The Economics of Business Culture: Game Theory, Transaction Costs and Economic Performance, Oxford: OUP, 1991. Schelling T. C., Altruism, meanness, and other potentially strategic behaviors, American Economic Review, 68(2): 229-231, May 1978. Crawford V. P., Thomas Schelling and the analysis of strategic behavior, in Strategy and Choice, ed. by R. J. Zeckhauser, MIT Press, 1991. For a history of game theory since Old Testament times, point your browser at the following URL: www.canterbury.ac.nz/econ/hist.htm www.pitt.eduhalroth/alroth.html Eddie Dekel, Drew Fudenberg and David K. Levine, Learning to Play Bayesian Games (Jun. 20, 2001). www7.kellogg.northwestern.edu/research/math/papers/1322.pdf www.gametheory.net/html/lectures.html Drew Fudenberg and David K. Levine, The Nash Threats Folk Theorem With Communication and Approximate Common Knowledge in Two Player Games (Jun. 10, 2002).