Nash Bargain Advisor

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Abstract

John Nash won the 1994 Nobel in Economics for his work on what came to be known as “Nash Equilibrium”, where two or more competing entities “cooperate” (without illegally colluding) to reach a “Nash Bargain”. A Nash Bargain is reached when two or more competitors produce optimal quantities of the same or similar product or service to maximize their own self- interest, assuming others are rational and will do the same. The book and movie “A Beautiful Mind” dramatized Nash’s life story and work.

A relatively simple Excel-based tool helps you calculate a Nash Bargain in a competitive situation. It is available for FREE.

What Can the Nash Bargain Advisor Do For You?

Whether you are in a competitive business situation or not, it is important to understand how producers and consumers may come to an effective, mutually-beneficial market solution. If a producer is selling a product or service in an “elastic” market, where demand increases with reduced price, profit may sometimes be maximized by reducing prices to increase sales.
 
The Nash Bargain Advisor can handle a competitive situation where you know: 1) The “Demand Curve” data for the market, namely the relationship between market price and quantity of product on the market, 2) Your own Cost Structure, namely the non-recurring investment to set up your production facilities and the recurring production cost of each item sold, and 3) An estimate of your competitor(s) Cost Structures.
 
The Nash Bargain Advisor will compute the optimal quantity, market price, and estimated profit (or loss) you are likely to make if you follow the given advice, and if your competitors independently do the same.
 

About “Elastic” Markets

Let us start with a monopoly, the simplest case. Say a given producer is the only source for some unique product or service. Of course, if it is “a necessity of life” they can charge anything they want for it. On the other hand, if people can do without it, the market will be elastic and the monopoly producer will have to set the price and quantity to obtain the highest profit.
 
It is a myth that the best way to increase profits is to increase prices. Often reducing prices will increase sales and reduce unit production costs such that overall profits increase. The figure below illustrates that case.
 

The heavy black line is the Demand Curve that indicates how the market price declines from about $12 per unit to $4 when the quantity on the market increases from 10 million to 100 million units. (You can change the Demand Curve by entering different numbers on the SETUP sheet of the Nash Bargain Advisor.)

 
The thin red and blue curves indicate the production Cost Structures per unit for two alternate production facilities, as a function of the number of units produced. (You can change the Cost Structures by entering different numbers on the SETUP sheet of the Nash Bargain Advisor.)
 
A producer (whether a monopoly or not) has to decide the optimum level of capital investment. Capital investment in more automated production facilities will increase initial, non-recurring costs, but may reduce incremental production costs by a sufficient amount to pay back the investment -or not- depending upon the number of units eventually sold and the market price when they are sold. The Nash Bargain Advisor allows you to enter and compare two different sets of Cost Structures.
 
In the graph above, the thin red curve represents a more highly-automated producer we’ll call “Alpha” and the thin blue dashed curve a less-automated alternative we’ll call “Beta”. Note that, if you produce a smaller number of units, the production cost for each (which is the recurring, incremental cost per unit plus the share of the non-recurring costs) will be higher than if you produce a larger number of units. A more automated facility, corresponding to higher non-recurring investment, will be at a relative disadvantage for low production quantities but may gain an advantage for larger production quantities, as indicated by the thin red and blue curves.
 
The heavy red and dashed blue curves indicate the profit per unit as a function of the number of units produced. You might think the maximum overall profit occurs when the profit per unit is maximized, but you would be wrong! The figure below illustrates the overall profit (or loss) for Alpha and Beta alternatives as a function of quantity produced. 
 
The overall profit for the alternative Cost Structures is maximized with a market quantity of 65 million for Alpha and 62 million for Beta, assuming each is a monopoly in a given market. This corresponds to a market price of $7.20 to $7.36 per unit. If the monopoly produces too few units, say 10 to 14 million, it will get $11.68 to $12 per unit, but will lose money overall. On the other hand, if produces too many units, say over 70 million, there will be a glut on the market and overall profits will go down substantially.
 
Different Demand Curve and production Cost Structures that you can experiment with on the SETUP sheet of the Nash Bargain Advisor may result in situations where overall profits increase or decrease monotonically as production quantities increase. However, it is much more typical for profits to maximize with a moderate number of units on the market and for there to be lower profits (or net losses) for very low or very high production quanities.  
 

About Competitive Markets

The insight John Nash brought to Economics and that gained him the Nobel in Economics for 1994 is that the situation is the same for multiple producers in a competitive marketplace. If two or more companies produce the same or similar products in an elastic market, such as Burger King and McDonalds or HP and Acer, it is to their advantage to collectively produce a certain number of units, neither too few nor too many.
 
If there are too few fast-food restaurants in a given geographic area, they may be able to charge a bit more per burger, but they will sell fewer as potential customers choose to eat at home or to go to full-service eateries. On the other hand, if there are too many fast-food places, they will have to reduce prices drastically to attract customers and their overall profits may decline or turn into losses.
 
The same is true for PC makers. As production quantities have multiplied, prices have come down sharply and features have improved dramatically. This, in turn, has increased sales to the point of nearly 100% market penetration in the US and other westernized countries. More and more people have at least one PC and some have a desktop plus a laptop, and other families have one for each member of the family. (My wife and I have one desktop plus three laptops between us.) With the economic slowdown, however, there may be too many units on the market and prices may drop to the point where some producers face losses and have to cut back production or drop out of the market.
 
So, how can competitors in an elastic market adjust production quantities such that they can each make a fair profit? Well, they could collude and fix quantities and prices and divide markets to increase their profits. However, that would be totally illegal!
 
Using game theory, John Nash came up with a way to reach “equilibrium” without illegal collusion. His solution is for each competitor to use their own Cost Structure and estimate the Cost Structures of competitors and calculate the quantity they should produce, assuming others are rational and will do the same. (The highlighted part of the previous sentence is the most important part. If competitors are not rational, or if they try to “cheat” by producing too many units, the Nash Bargain will not work.)
 
The Nash Bargain Advisor calculates the optimal quanities each competitor should produce to maximize their own self-interest, assuming others “cooperate” by doing the same in a rational way. The Nash Bargain Advisor also calculates the consequences if one or more producers “cheat” and over-produce more than their optimal quanitiy, or, if one or more producers under-produce due to miscalculation or disruption in supplies or production facilities.
 

How to Use the Nash Bargain Advisor

You have to enter only eight items of data. The SETUP sheet is shown below.
 

The first four entries define the Demand Curve. In the real world, that data would come from marketing surveys or actual experience with sales volume at various outlets with different prices. It is assumed we are working in the relatively linear portion of the Demand Curve, where market price and quantity available vary inversely.

 
Enter the minimum reasonable number of units that may be on the market in the first cell. In the above example, that number is 10 million units. With that number of units available, market demand will support a price of about $12 per unit, entered into the second cell.
 
Then, enter the maximum reasonable number of units and the estimated price market demand will support. In the example, 100 million units will drive the price down to about $4 per unit.
 
The next four entries define the Cost Ctructures for two different, competitive producers that we will call Alpha and Beta.
 
The first cell for each producer contains the non-recurring costs for setting up the production facility and various fixed costs that do not vary with production volume. In this case, Alpha is assumed to have invested $150 million dollars and Beta $120 million.
 
The second cell for each producer contains the incremental production cost per unit, including supplies, factory, distribution, and sales costs. Alpha is assumed to produce units at $1.40 each and Beta at $1.90.
 
For this example, the Nash Bargain Advisor calculates that Alpha should optimally produce about 32 million units and Beta about 31 million, for a total of about 63 million units on the market.
 
The graphs on the SETUP sheet indicate the options Alpha and Beta have, assuming their competitor “cooperates” with the Nash Bargain. (A larger version of these graphs is available on the COOPERATION sheet.) The figure below shows the situation for Beta asuming Alpha produces their Nash Bargain quantity.
 

Note that Beta could increase their profits by producing about 40 million units, about 9 million more than the Nash Bargain calls for. However, doing so will reduce Alpha’s profits to nearly zero, making Alpha, in game theory terms, the “SUCKER”. That would most likely prompt Alpha to retalliate by also over-producing in the next production cycle. If Beta produces fewer than the Nash Bargain quantity, their profits will decrease. Beta will lose money if they decrease below about 24 million units. If Beta under-produces, Alpha, the cooperator, will see the market price go up and Alpha will earn greater profits.

 
The situation is similar for Alpha, assuming Beta produces their Nash Bargain quantity, see the figure below.
 

So, for their long-term self-interest, both Alpha and Beta should refrain from cheating and avoid getting into a “price war”.

 
SOME SPECIFIC EXAMPLES OF COOPERATION AND OVER- OR UNDER-PRODUCTION
 
The Nash Bargain Advisor also calculates nine examples for the cases where both Alpha and Beta cooperate and where one or both competitors over- or under-produce.
 
Here is a summary based on the SETUP data discussed in the previous paragraphs:
 
  • BOTH COOPERATE: Market price is $7.24, Alpha makes a net profit of $40M and Beta $46M.
  • ALPHA CHEATS (over-produces by 50%): Market price drops to $5.80, Alpha (Cheater) makes a hefty net profit of $64M and Beta (the SUCKER) is driven down to a profit of $1M.
  • BETA CHEATS (over-produces by 50%): Market price drops to $5.86, Beta (Cheater) makes a hefty net profit of $65M and Alpha (the SUCKER) is driven down to a loss of $5M
  • BOTH CHEAT: Market Price drops to $4.42, both Alpha and Beta lose about $3M each.
  • ALPHA UNDER-PRODUCES (by 50%): Market price is driven up to $8.68. Beta (Cooperator) makes a hefty profit of $91M and Alpha (under-producer) takes a loss of $32M.
  • BETA UNDER-PRODUCES (by 50%): Market price is driven up to $8.62. Alpha (Cooperator) makes a hefty profit of $84M and Beta (under-producer) takes a loss of $15M.
  • BOTH UNDER-PRODUCE (by 50%): Market price is driven up to $10.06. Alpha takes a loss of $9M and Beta’s profits go down to only $7M.
  • ALPHA CHEATS (over-produces by 50%) and BETA UNDER-PRODUCES (by 50%): Market price is $7.18, Alpha (Cheater) makes a hefty profit of $130M and Beta (under-producer) takes a loss of $31M.
  • BETA CHEATS (over-produces by 50%) and ALPHA UNDER-PRODUCES (by 50%): Market price is $7.30, Beta (Cheater) makes a hefty profit of $132M and Alpha (under-producer) takes a loss of $54M.
 
Cheating by one or more competitors gluts the market with excess product and drives market price down. That, at least temporarily, benefits consumers. If a price war develops, consumers may benefit for several years. However, there is a risk that one or more competitors may be driven out of business by losses sustained in a price war and that may leave the market to one supplier, which may raise prices for the consumer.
 
If one or more competitors under-produces, that hurts consumers by reducing supplies below market needs and raising consumer prices.
 
If some of the competitors under-produce while others over-produce such that the net production quantity on the market is around the number called for by the Nash bargain, that will neither hurt nor benefit the consumers. However, the under-producer will see a drop in profit and perhaps endure a loss, while the over-producer will see a hefty profit. Therefore, if one or more competitors is hit by a disruption in production, the best action is for the other competitor(s) to over-produce to make up for the disruption..
 
Therefore, it appears that the best long-term situation for consumers and producers is a competitive market where producers meet their Nash Bargain quantities and do not cheat. Consumers benefit from reasonable and relatively stable prices while producers make a fair profit.
 

How to Use Nash Bargain Advisor for More Than Two Producers

The Nash Bargain Advisor may be used for more than two producers by combining additional producers into Alpha or Beta. For example, say there are four producers in a given market and two have advanced production facilities while the other two basic facilities. You could use Alpha for the two advanced facility and Beta for the two basic facilities, combining their non-recurring costs and dividing their Nash Bargain quantities.