In the social sciences, bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service debate the price or nature of a transaction. If the bargaining produces agreement on terms, the transaction takes place. It is often commonplace in poorer countries, or poorer localities within any specific country. Haggling can mostly be seen within street markets worldwide, wherein there remains no guarantee of the origin and authenticity of available products. Many people attribute it as a skill, but there remains no guarantee that the price put forth by the buyer would be acknowledged by the seller, resulting in losses of profit and even turnover in some cases. A growth in the country's GDP Per Capita Income is bound to reduce both the ill-effects of bargaining and the unscrupulous practices undertaken by vendors at street markets.
Although the most apparent aspect of bargaining in markets is as an alternative pricing strategy to fixed prices, it can also include making arrangements for credit or bulk purchasing, as well as serving as an important method of clienteling.
Bargaining has largely disappeared in parts of the world where retail stores with fixed prices are the most common place to purchase goods. However, for expensive goods such as homes, antiques and collectibles, jewellery and automobiles, bargaining can remain commonplace.
Dickering and "haggling" refer to the same process.
Bargaining, also called haggling, is common in many countries, such as most of Asia, Latin America and North Africa. In other places, it may generally be used only for large items with no fixed price, such as buying a house or a used car. However, bargaining is possible in almost any flea market or tourist shop anywhere. In such places, if you don't bargain, you are almost certain to pay more than necessary. Vendors expect a bit of bargaining so their initial asking price is considerably higher than they hope to receive, which in turn is higher than the minimum they could accept and still make a profit.
This article is about cultures where bargaining is common. In some other places it may be possible to ask for a discount, but there the nominal price is close to what you will pay and your first suggestion should be reasonable, perhaps the price you hope to get.
By all means bargain hard and try not to get cheated, but do not expect too much. You are a visiting amateur going up against a professional on their home turf. Just holding your own and getting a reasonable price will be an accomplishment; do not expect to achieve some miraculously low price. Against a pro boxer, almost anyone would be justifiably proud just to leave the ring on their feet; hoping to win the bout would be foolish and hoping to score a knockout utterly ludicrous.
Don't get upset if you pay somewhat more than a local would; that is quite normal in many areas. Even getting "cheated" in a local bazaar, perhaps paying $25 for something a local could buy for $10, is often better than buying it in some overpriced airport shop. It will usually be cheaper if you have even basic bargaining skill, and buying in the bazaar puts money into the local economy, rather than giving it to some large company or, in some places, helping crooked officials line their pockets.
If possible, try to buy in areas where there are many vendors and competition may bring prices down. In an example from China, one traveller reports finding some lovely silk shawls in the only store at a well-known scenic site, beating the price down down from ¥250 to ¥100, and thinking he had done reasonably well. A few days later he found the same shawls in a multi-vendor commercial area with an asking price of ¥80 and bought a half dozen at ¥55 each. He felt a bit foolish but did not feel he had been robbed since back home those shawls would likely have cost at least the equivalent of ¥350, and quite likely twice that; even a naive tourist paying ¥200 is not being robbed.
In markets where haggling is the norm, the rhythm of the exchange is as much a social dance as a price talk. A polite smile, a measured pause, and the willingness to let the seller suggest a lower figure signal respect for the vendor’s craft. Avoiding blunt rejections—"no, that's too high"—in favour of a lighter "perhaps a little less?" keeps the interaction friendly and often yields a better deal. Locals will notice the tone of your voice as much as the numbers you quote; a courteous approach can open the door to informal discounts that aren't advertised at all.
Seasonal festivals and public holidays can dramatically alter the bargaining landscape. During peak tourist periods vendors may raise base prices, anticipating that visitors will accept higher offers. Conversely, in the days following a major local celebration, stalls often clear stock at reduced rates to make room for new wares. Savvy shoppers time their visits to these lull periods, when sellers are more motivated to finalise a sale than to parade premium price tags. Observing the flow of foot traffic and noting when the crowd thins can be as valuable as any negotiation skill.
Consider your priorities. If you are bargaining over some silk item that would cost $200 back home, it may not be worth worrying about whether you pay $20 or $25 in Thailand. If you make good money and are spending a substantial chunk of it on a trip, then it makes no sense to waste half an hour to save $5; your time is worth more than that and you have plenty of better things to do with it during a trip.
Also consider the vendor's situation. Sometimes the amounts you are arguing over are just a pittance to a traveller from a relatively rich country, but are quite important to someone in a poorer country who needs to make a living off the tourist trade. An amount that is large to you may be huge to the vendor. In some cases, for example, a big sale on an expensive item such as a carpet might mean the difference between all the vendor's children going to school this year or just the boys.
See the Shopping article for some alternatives that may let you avoid bargaining.
The key to making a good deal is knowing the right price. If you know the right price you can just state your price, start leaving the store and your offer will usually be accepted. Try to have a rough understanding of the item's value before you start haggling.
Hotel or airport gift shops generally have (high) fixed prices that will at least give you an upper boundary.
Government-run craft shops exist in many countries to market local products. These also often have fixed prices, but more reasonable ones that give a more useful boundary. They also generally have good quality so for many travellers buying in them is a sensible alternative to haggling elsewhere.
Shop around, especially for items available from multiple vendors in tourist areas.
Ask other travellers what they paid for similar goods and try to make a better deal.
If you buy several similar items, try to make a better deal each time.
If you can, ask a trusted local what price range is appropriate.
If possible, try to see what locals pay and refuse to pay more. You can either watch, or ask someone after they have purchased. Try not to be obvious; in some parts of the world locals will defer purchases so that the vendor can extract additional profits from you as an outsider.
Do not let unknown locals "help", with either bargaining or finding what you need; you will very likely end up paying an extra commission. In many places, this includes tourist guides and taxi or rickshaw drivers; some shops pay them substantial commissions to bring in customers, those shops are usually overpriced, and some guides or drivers will take you only to such shops. To get good prices, you need to go shopping without a guide and preferably on foot.
It may help if you can adjust the vendor's expectations. If he thinks you are rich, the price may go up. To get a better price, tell him you are a backpacker on a tiny budget, a student, an English teacher on a low local salary, or whatever. You cannot really afford what he is selling, but you like it a lot; can he do something to make it possible? When doing this, try not to look rich; a Rolex on your wrist and $2000 worth of camera around your neck will definitely not help.
Be strong. Don't let them get to you, no matter how hard they push. You might be offered tea, coffee, snacks, etc. You can accept them and it does not mean you have to buy anything, although you may be 'guilt-tripped' later. You have only one obligation; to buy once a price is agreed. Until you make an offer that the vendor accepts or vice versa, you have absolutely no obligations beyond common courtesy.
During the actual bargaining:
Be courteous and friendly (but firm) in your negotiations. If the vendor takes a personal liking to you, you will almost always get a better deal.
Just as vendors often start with absurdly high prices, in most cases you should start with a low initial offer. For example, if you think an item is worth around $100 and he or she asks $500, offer $20. This gives you some negotiating room. More generally, your first offer should always be well below what you hope to pay. Don't go overboard though: offering a dollar for something worth $500 wastes everyone's time and indicates to the vendor that further talk with you is not likely to be worthwhile.
Try to move in small increments. For example, if you think $500 might be a fair price, the vendor asks $1000, you offer $200 and he counters at $800, resist the temptation to move quickly toward a deal by offering $400. Offer $250 which indicates you are willing to negotiate but puts pressure on him to come down more if he wants a deal.
Later in the process, use even smaller increments; if you think $500 might be fair, your last offer was $400 and he says $600, don't jump to $500 or even $450. Either tell him $400 is the best you can do or offer something like $410 or $420.
A common tactic is to bid the vendor farewell and start walking off. You will usually get at least two offers as you walk away, each lower than the previous. Alternatively, the vendor may ask "How much do you want this?" (or words to that effect), which acknowledges the fact that they realise a potential sale is walking out of the door. In some cultures it is also common for the salesman to walk off if the price is too low. This occurs the closer you get to the profit threshold but stay firm, the salesman will return in minutes to try more bargaining (Ex: Ghana beach salesman).
You really should walk away from bad deals, especially for tourist items where there are many vendors in an area. If someone asks $150 for something worth about $10, don't even try to bargain; just look for another vendor.
Some guides and articles suggest that, when you have no idea what a fair price might be, you should offer a fixed percentage (a half, a third, a quarter...) of the shopkeeper's first price. Alas, in general this doesn't work: many shopkeepers are perfectly aware of this tactic and will thus first offer an absolutely ridiculous price that can be tens or hundreds of times more than the real value, which they will then be more than willing to negotiate down to "half".
If you feel that you must have a general rule to go by in those situations, remember that in tourist areas it is quite common for a moderately skilled bargainer to pay below a quarter of the initial asking price, and an eighth is not unheard of. You should start below what you hope to pay, so your first offer should be around a tenth of the asking price. If you really want to just get the bargaining over with, and do not mind paying over the odds to accomplish that, you could go as high as a fifth.
The more you know about the merchandise, the better. For example, you can buy cheap pottery in any flea market on Earth with no expertise at all, but if you are planning a trip, want to spend a significant amount to bring home some fine porcelain, and are not an expert, then you should read a few books on the subject and visit museums (either near home or at the destination) before shopping. The Wikipedia articles on the topic may be a fine starting point. See our articles on carpets or gemstones if you want those.
Provided you understand the merchandise, it is often worth paying for quality. For example, a $200 pair of boots—skillfully made of fine leather and likely to last for years—may be a much better buy than a $20 pair that may fall apart in weeks.
If you do not understand the merchandise well, avoid expensive items. In the example just given, the risk is that you may pay $200 for boots that would not be a good buy at $20.
The more expensive the item, the more important this is; there are forgeries of all sorts of expensive items, from Stradivarius violins to Nikon lenses, and modern replicas of many expensive antiques. Replicas can be a good buy at the right price, but not at the cost of an original.
The same advice holds for items that come in a range of types or grades at widely varying prices. A fine carpet or a topnotch wine may be a good buy even at a high price, and a cheap one worthwhile even if its quality is relatively low. However, if you don't understand the differences, then you should only consider the cheaper ones.
The more you know about the country you are in, the better. You may not get the price a local would pay, but you are almost certain both to enjoy your shopping more that an ignorant foreigner would and to get a better price. Specifically:
If you are in a country that does not use Western numerals, then learn the local numbers. It will save you a lot of time and money when you are bargaining about a hotel room and there is a price list right in front of you. You should still bargain, but it gives you a starting point. Sometimes having a piece of paper and pen or a calculator to display the price you're offering can be helpful.
Even better, learn some of the local language. Even just being able to count in the local language can win you some respect and therefore a better price. If you can, stick to the local language even if the seller uses English or your own language. Some vendors outright say that they offer better deals if you speak the local language and speaking enough of the local language enables you to understand what the vendor says to her colleagues or other customers.
Investigate local superstitions (see the Buy sections of country articles for the ones we know of) and use them if possible. In some countries, the first deal of the day or of the new year is thought to indicate how the rest of the day or year will go for the merchant. If a vendor believes that, then he or she really does not want to lose that sale, so being there at the right time can be to your advantage.
Some bargaining techniques take more effort or planning, more nerve, more practice, or a bit of acting skill. These may even be worth practicing beforehand.
If there are two or more of you, you can wax theatrical. He wants the item, but she holds the purse strings and won't pay the price, or vice versa. Or—perhaps easy for some couples to act—she wants to continue shopping, but he is bored and just wants to go find lunch or beer. Consider the "good cop/bad cop" tactics from a hundred movies.
As a variant of that, ethnically mixed groups can sometimes turn the common guide's commission system to their advantage. Consider a British husband and Chinese wife meandering about a Chinese market together. What should she do when a vendor assumes she is a tourist guide and offers her a commission to act as translator, and as accomplice in cheating the foreigner? Her best course may be to accept, surprise the vendor by bargaining hard, and later collect her commission to bring the cost down further. Consider the tactics of a "double agent" deceiving one employer in the interests of the other.
If bargaining for something unique or expensive, such as a good carpet, don't show too much interest in the item you actually want or the vendor will know that's your only choice and price accordingly. Take your time and browse other items. Consider the tactics of a woman "playing hard to get", which she may do even with a man she actually considers fascinating.
As a variant of that, assure the vendor there is no rush. Encourage him to go off a deal with another customer while you browse, or tell him you don't plan to buy today; you'll just look in several stores and come back next week to actually purchase. This puts more pressure on him if he wants the sale today, or wants to sell to you before you visit his competitors.
In some cases, you can tell the vendor you'd like to buy, but it's illegal. There are strict international rules on the trade in ivory and products from endangered species, and many nations restrict export of antiques; for example China forbids export of anything older than 1911. If the vendor then admits that the "ivory" is fake or the expensive "Tang Dynasty" vase is actually a modern reproduction, you can use that information in further bargaining. If not, you can moan about the legal risk in an effort to drive the price down.
If the vendor's asking price is absurdly high, by all means laugh or show astonishment in some way. Ideally, ask in the local language if he is crazy or joking. This is not considered rude, provided you smile as you say it, and it indicates to the vendor that you are aware of the item's real value, even if you are not.
When bargaining, do so responsibly.
Even in cultures where haggling is the norm, many items do have fixed prices. For example, groceries and alcohol usually have fixed prices. Do not haggle when buying e.g. bus tickets; check for a price list in the bus terminal or ask the other passengers in the line or look over the shoulder of the one in front of you to see what the locals pay. Confusingly, items in the same category may have a fixed price in one place and be negotiable in another – sometimes within the same country. For example, while a taxi fare inside Estelí will have a fixed price, if you want to get a cab in Managua you can haggle and must agree on a price before getting aboard.
Choose your battles. By all means bargain when buying a carpet from a posh bazaar shop. But if a bottle of water is too expensive, buy it somewhere else.
Don't back out. If you make an offer consider yourself committed to buy if the vendor accepts that price. Don't waste your time or the seller's time bargaining if you have no intention of buying. Even an offer made in jest creates an obligation. As you walk by, the vendor waves a sword worth ¥200 at most and asks for 800. You laugh, say "maybe 100" and walk on. If he calls you back yelling "OK", give him the 100 and take the sword.
Do not let the other person "lose face". Often it is said that "everything is negotiable"—but it isn't. Loss of face is never negotiable. Be aware that the person with whom you are dealing has a family and responsibilities. You are trying to find an agreed position.
Above all, don't take it too seriously; have a sense of humour and know when to accept an offer. Remember that vendors are generally not evil swindlers attempting to trick people out of their hard-earned money; they are often just business people working to support their families.
Game theory is a group of mathematical techniques that can be used to analyze almost any human interaction, not just things that are actually games. It has a fairly wide range of applications in both psychology and economics. In particular, it can be applied to almost any situation involving negotiation or bargaining, and it has been; in fact there is a whole branch of the theory that deals with "bargaining games".
A basic part of the theory is a distinction between different classes of game:
In a zero-sum game, the total value at stake does not change:
Five poker players who each bring $1000 to the table. Some players may have a profit and others a loss, but the sum of all profits and losses is always zero. No matter how the game goes, there will still be a total of $5000 at the end.
Chess at 1 point for a win, zero for a loss and half for a draw. Total score, for any number of players and rounds, always equals the number of games, whatever happens on the board.
In a non-zero-sum game, the total value involved changes:
a good marriage can be a fine thing for both partners (a "win-win" situation)
a bad marriage can make both partners exceedingly miserable
some business deals are good for both companies
some wars damage both nations without producing any noticeable positive effects ("lose-lose"?)
A mixed game has both zero-sum and non-zero-sum components:
The scoring for chess is zero-sum, but other parts of the situation are non-zero-sum; both players may be paid to compete and both may be exhausted after a match.
The scoring for a boxing match is zero-sum, but both fighters may be injured.
Bargaining is a mixed game.
If you pay $70 instead of $50 for an item, the vendor gets $20 more and you have $20 less in your pocket; this is clearly zero-sum.
Overall, the goal is to reach a deal that is reasonable for both players; the vendor gets a sale and you get the item. This is a classic win-win outcome, clearly non-zero-sum.
The two goals are basically in conflict; reaching a deal may require the buyer paying more or the vendor accepting less. Sometimes the conflict is not reconcilable without a major sacrifice by one player. In those situations, the best solution is to walk away. You should not want to make such a sacrifice yourself and cannot expect one from the vendor.
When haggling, your goal should not be to "win", or to get the lowest possible price, or to eliminate the vendor's profit, but to find a mutually satisfactory price.
Art and antiques shopping
Budget travel
Carpets
Experienced buyers understand that bargaining is not solely about lowering cost; it is also a chance to shape the transaction's ancillary terms. Asking for a bundle—adding a related accessory or securing a longer warranty—can be more rewarding than a marginal discount. In many street markets, vendors welcome small add‑ons as a sign of goodwill, especially when the buyer shows genuine interest in the product's story or origin. Framing your request around the value you place on the item, rather than just the price, often encourages sellers to accommodate your terms.
Negotiation between a buyer and seller over the price and nature of their transaction For other uses, see Bargaining (disambiguation). "Dicker" and "Dickering" redirect here. For other uses, see Dicker (disambiguation). This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Bargaining" – news · newspapers · books · scholar · JSTOR (August 2018) (Learn how and when to remove this message) People bargaining in a traditional Indonesian pasar malam (night market) in Rawasari, Central Jakarta. In the social sciences, bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service debate the price or nature of a transaction. If the bargaining produces agreement on terms, the transaction takes place. It is often commonplace in poorer countries, or poorer localities within any specific country. Haggling can mostly be seen within street markets worldwide, wherein there remains no guarantee of the origin and authenticity of available products. Many people attribute it as a skill, but there remains no guarantee that the price put forth by the buyer would be acknowledged by the seller, resulting in losses of profit and even turnover in some cases. A growth in the country's GDP Per Capita Income is bound to reduce both the ill-effects of bargaining and the unscrupulous practices undertaken by vendors at street markets. Although the most apparent aspect of bargaining in markets is as an alternative pricing strategy to fixed prices, it can also include making arrangements for credit or bulk purchasing, as well as serving as an important method of clienteling. Bargaining has largely disappeared in parts of the world where retail stores with fixed prices are the most common place to purchase goods. However, for expensive goods such as homes, antiques and collectibles, jewellery and automobiles, bargaining can remain commonplace. Dickering and "haggling" refer to the same process.
[edit] Dongxiangs bargaining for sheep in China's Gansu province. Haggling is associated commonly with bazaars and other markets where centralized regulation is difficult or impossible. Both religious beliefs and regional custom may determine whether or not the sellers or buyers are willing to bargain. Regional differences[edit] Spice market Djerba, Tunisia. In North America, Australia, and Europe, bargaining is typically confined to transactions involving expensive or one-of-a-kind items (automobiles, antiques, jewelry, art, real estate, trade sales of businesses) and informal sales settings such as flea markets and garage sales. In other regions of the world, bargaining may be the norm, even for small commercial transactions. In Indonesia and elsewhere in Asia, locals haggle for goods and services everywhere from street markets to hotels. Even children learn to haggle from a young age. Participating in that tradition can make foreigners feel accepted. On the other hand, in Thailand, haggling seems to be softer than the other countries due to Thai culture, in which people tend to be humble and avoiding of arguments. However, haggling for food items is strongly discouraged in Southeast Asia and is considered an insult, because food is seen as a common necessity that is not to be treated as a tradable good. In almost all large complex business negotiations, a certain amount of bargaining takes place. One simplified 'western' way to decide when it's time to bargain is to break negotiation into two stages: creating value and claiming value. Claiming value is another phrase for bargaining. Many cultures take offense when they perceive the other side as having started bargaining too soon. This offense is usually as a result of their wanting to first create value for longer before they bargain together. The Chinese culture, by contrast, places a much higher value on taking time to build a business relationship before starting to create value or bargain. Not understanding when to start bargaining has ruined many an otherwise positive business negotiation. In areas where bargaining at the retail level is common, the option to bargain often depends on the presence of the store's owner. A chain store managed by clerks is more likely to use fixed pricing than an independent store managed by an owner or one of the owner's trusted employees.[citation needed]
[edit] Bargaining is not just a rational negotiation process; psychological factors play a crucial role in its outcomes. Research suggests that emotions, cognitive biases, and perceived fairness significantly influence bargaining behavior. For example, studies have shown that individuals experiencing anger during negotiations tend to make more aggressive offers, while those feeling guilt or regret are more likely to make concessions (Thompson et al., 2010). Additionally, the endowment effect—a cognitive bias where people overvalue items they own—can impact bargaining outcomes by making sellers demand higher prices than buyers are willing to pay. These psychological influences highlight the complexity of bargaining beyond economic models. Thompson, L., Wang, J., & Gunia, B. C. (2010). "Negotiation." Annual Review of Psychology, 61, 491-515.
[edit] Medina Tripoli, Libya Behavioral theory[edit] The personality theory in bargaining emphasizes that the type of personalities determine the bargaining process and its outcome. A popular behavioral theory deals with a distinction between hard-liners and soft-liners. Various research papers refer to hard-liners as warriors, while soft-liners are shopkeepers. It varies from region to region. Bargaining may take place more in rural and semi-urban areas than in a metro city.[citation needed] Game theory[edit] Bargaining games refer to situations where two or more players must reach an agreement regarding how to distribute an object or monetary amount. Each player prefers to reach an agreement in these games, rather than abstain from doing so. However, each prefers that the agreement favor their interests. Examples of such situations include the bargaining involved in a labor union and the directors of a company negotiating wage increases, the dispute between two communities about the distribution of a common territory, or the conditions under which two countries agree on nuclear disarmament. Analyzing these kinds of problems looks for a solution that specifies which component in dispute corresponds to each party involved. Players in a bargaining problem can bargain for the objective as a whole at a precise moment in time. The problem can also be divided so that parts of the whole objective become subject to bargaining during different stages. In a classical bargaining problem, the result is an agreement reached between all interested parties or the status quo of the problem. It is clear that studying how individual parties make their decisions is insufficient for predicting what agreement will be reached. However, classical bargaining theory assumes that each participant in a bargaining process will choose between possible agreements, following the conduct predicted by the rational choice model. It is particularly assumed that each player's preferences regarding the possible agreements can be represented by a von Neumann–Morgenstern utility theorem function. Nash defines a classical bargaining problem as being a set of joint allocations of utility, some of which correspond to what the players would obtain if they reach an agreement, and another that represents what they would get if they failed to do so. A bargaining game for two players is defined as a pair (F,d) where F is the set of possible joint utility allocations (possible agreements), and d is the disagreement point. For the definition of a specific bargaining solution, it is usual to follow Nash's proposal, setting out the axioms this solution should satisfy. Some of the most frequent axioms used in the building of bargaining solutions are efficiency, symmetry, independence of irrelevant alternatives, scalar invariance, monotonicity, etc. The Nash bargaining solution is the bargaining solution that maximizes the product of an agent's utilities on the bargaining set. The Nash bargaining solution, however, only deals with the simplest structure of bargaining. It is not dynamic (failing to deal with how Pareto outcomes are achieved). Instead, for situations where the structure of the bargaining game is important, a more mainstream game-theoretic approach is useful. This can allow players' preferences over time and risk to be incorporated into the solution of bargaining games. It can also show how the details can matter. For example, the Nash bargaining solution for the prisoners' dilemma is different from the Nash equilibrium. Bargaining and posted prices in retail markets[edit] Retailers can choose to sell at posted prices or allow bargaining: selling at a public posted price commits the retailer not to exploit buyers once they enter the retail store, making the store more attractive to potential customers, while a bargaining strategy has the advantage that it allows the retailer to price discriminate between different types of customer. In some markets, such as those for automobiles and expensive electronic goods, firms post prices but are open to haggling with consumers. When the proportion of haggling consumers goes up, prices tend to rise. Processual theory[edit] This theory isolates distinctive elements of the bargaining chronology in order to better understand the complexity of the negotiating process. Several key features of the processual theory include: Bargaining range Critical risk Security point Integrative theory[edit] Integrative bargaining (also called "interest-based bargaining," "win-win bargaining") is a negotiation strategy in which parties collaborate to find a "win-win" solution to their dispute. This strategy focuses on developing mutually beneficial agreements based on the interests of the disputants. Interests include the needs, desires, concerns, and fears important to each side. They are the underlying reasons why people become involved in a conflict. "Integrative refers to the potential for the parties' interests to be [combined] in ways that create joint value or enlarge the pie." Potential for integration only exists when there are multiple issues involved in the negotiation. This is because the parties must be able to make trade-offs across issues in order for both sides to be satisfied with the outcome. Automated bargaining[edit] When a bargaining situation is complex, finding Nash equilibrium is difficult using game theory. Evolutionary computation methods have been designed for automated bargaining, and demonstrated efficient and effective for approximating Nash equilibrium.