Dynamic Delay Bargaining Protocol

In the previous section we have shown the benefit of allowing non-monotonic offers in avoiding the cost from the delay generated from establishing a new bargaining session, in which we assume a decreasing valuation of the buyer (see Figure 4-1). In this section we analyze other bargaining situations. Under these situations, we will show the benefit of allowing a buyer to delay their offer without assuming a decreasing valuation over time.

4.2.1 Motivation

One of the advantages of artificial agents is their fast response, resulting in a short negotiation session. For example, a bargaining session between two agents may conclude within seconds or even milliseconds. Generally, when a negotiation needs no complex information processing, efficiency can be gained by increasing the speed of agent responses. However, in some cases, faster resolution may not be the ultimate goal of the negotiation. Consider the following example.

Example 4-2. Suppose a buying agent is delegated to buy a product within three days. Then a deal made on the first day is no different from a deal on the third day; but finding a lower price and/or better quality item is a priority. If the buyer does not have

a cheaper price, then it may act cautiously on the first day, i.e. assigning a relatively low qt and/or Bt, and/or a relatively high pt(x) and/or EG’t+1, which causes a very slow concession rate by the buyer. As time goes by, the buyer will act more aggressively, i.e.

assigning a higher qt and/or Bt, and/or a relatively high pt(x) and/or EG’_t+1, which speeds up its concession rate.

In Example 4-2 above, the ideal strategy of the buyer is to open negotiation with all possible sellers within three days and accept the global best offer from the sellers (e.g.

the lowest price that those sellers are willing to sell for before the buyer’s deadline passes). Unfortunately, this strategy may not work because the seller who offers the global lowest price may leave the negotiation whilst the buyer is still bargaining with other seller(s), or a seller may accept a buyer’s counter-offer which is higher than the global best offer. This situation may happen because the buyer does not know whether or not a local best offer (i.e. the best offer in a given time interval) is globally best, and it cannot predict what will be the global best offer. Thus, the best strategy of the buyer is to concede slowly and keep its negotiation open with preferred seller(s), who provide local best offers, until it is convinced that that a local best offer is globally best. This option of keeping a negotiation open is known as a strategic delay. In general, any attempt to prolong a negotiation by a deliberate slow concession, or by taking no action within an allowed time, or, by submitting a message such as “please wait…” or irrelevant/meaningless messages, is considered as a strategic delay.

People use strategic delay in both complete and incomplete information games [Roth et al., 1988]. Delay is also observed in both finite and infinite horizon games.

Game-theoretic work on strategic delay has focused on finite horizon games (bargaining

with a deadline), including those with complete yet imperfect information [Dekel, 1990], complete information with inadvertent random delay [Ma and Manove, 1993], complete information with increasing surplus [Larson and Sandholm, 2002], incomplete information of agents’ valuation [Cho, 1990; Gu and Kuhn, 1998], stochastic agents’

deadline [Sandholm and Vulkan, 1999; Yildiz, 2004], etc. In the infinite horizon game context, most studies concern incomplete information cases with discount rate or fixed cost, such as two-sided uncertain valuations [Cramton, 1992], etc. Nevertheless, most of the work above is from a game-theoretic perspective. Even when an agent decision function is prescribed, it is only valid under very restrictive assumptions and bargaining settings, such as both bargainers are rational with unlimited computational power, with perfect foresight to solve the game, without risk of breakdown from a strategic delay, etc.

These assumptions are less realistic in open multi-agent systems. In contrast to game-theoretic work, our goals here are to show the importance of allowing delay in automated negotiation in open systems, to propose an agent’s decision structure when using a delaying strategy, and to demonstrate the costs and benefits of delay by naïve (myopic) agents.

4.2.2 Strategic Delay in Automated Negotiation

As in our previous analysis, we assume that agents can depict all negotiated issues into a single real number (e.g. utility value) and are able to make choices and their offers according to this value. However, in some cases, agents may not have exact properties but only estimates, e.g. estimated reservation price or a fuzzy time deadline. Also, we assume that an agent can bilaterally negotiate with multiple opponents simultaneously.

Example 4-3. Consider an e-market in which buying and selling agents can bargain over a product. Since both agents represent different users with different valuations, they will not reveal their private information. Besides, identical/similar items may be dynamically available from multiple sellers; thus, a buyer may simultaneously bargain with several sellers and prolong its negotiation with a preferred seller until it is convinced that no more sellers can provide a better offer.

Example 4-4. Similar to Example 4-3 except that the price of the negotiated item is decreasing over time, e.g. as in electronics. Suppose a potential buyer is not in a hurry to buy the item. Then, his/her agent should prolong the negotiation until the deadline is approaching because by that time the market price may be lower and, thus, it can ask for a lower price.

Example 4-5. Similar to Example 4-3 except that users have a vague valuation toward the item, for example, in a P2P system in which virtual currency is used to buy/sell digital items. When similar items are observed from many sources, users may change their preferences very frequently, especially at the beginning of the negotiation when the users have not made a definite decision. Thus, they may change the valuation fed to their agents and their agents may use strategic delay until the valuation is stable.

The first reason for delay is that overpricing may occur if the valuation is reduced after the transaction is made. For example, let the user’s initial valuation be $100 and a minute later becomes $80. If within few seconds his/her buying agent has bought the item for $90, then an overpricing occurs. The second reason for delay is that a missed-opportunity may happen if the valuation is raised after the agent leaves a bargaining session due to its initial low valuation.

Example 4-6. Consider an e-service in which a client may rent a scarce resource from a server within a specific time, e.g. doing a data analysis after all data have been collected. Since the complexity of the service varies for different cases, the renting price could be bargained and could depend on the service/renting duration. If the client does not have an exact schedule to do the analysis (e.g. the data collection can only be stochastically determined) but it wants to analyze the data soon after they are available, then it may delay the transaction until the data are ready for a shorter renting time (thus, a lower cost).

Example 4-7. An argumentation-based negotiation in which agents need time to verify arguments from their opponent. Suppose a seller claims that “My price is very cheap compared to others’ prices.” For its rebuttal, the buyer may verify the seller’s claim by checking any available past-transaction information, which may need a certain time. If the buyer does not have any other counter-argument but does not want to accept the seller’s offer immediately, then a strategic delay can be used until the verification is completed.

We can see from Examples 4-3 to 4-7 that the benefit of delay can be in the form of direct utility (e.g. paying a lower price in Example 4-3), and/or indirect utility (e.g. a more certain market price in Example 4-7). Generally, agents will use strategic delay if

• they have increasing valuation/surplus over time;

• they are averse to ambiguity/vagueness (imprecise probability/outcomes) that is reducing over time;¹ and

• they are willing to wait for a better opportunity.

In conclusion, there are many applications in automated negotiation in which agents may use strategic delay. Intuitively, a delay may increase the utility of agents, but it may deteriorate joint utility. In the next sections we will formally analyze the benefit from delay.

4.2.3 Basic Model

The following assumptions are used in our model.

Assumption 4-4. (Concurrent bilateral negotiation) A buyer may bargain with a set of sellers {S₁, S₂, …, S_n}, but all bargaining sessions are independent and asynchronous in the sense that the decision by two sellers are independently made (no collusion among sellers).

Assumption 4-5. (Persuasive negotiation) An agent may persuade their opponent