AM04 Duopoly

Main Page for Course:  Advanced Microeconomics: PIDE, Sep ’17

Lecture 4: on Duopolies — Simple Models contradict main methodological principles and conclusions of mainstream conventional textbooks

Link to Video (90m) of Lecture: AM04 Duopoly


Varian start his intermediate micro text by stating the maximization and equilbrium are the core principles of micro. Krugman recently stated that I am a “maximization and equilibrium” kind of guy. The goal of this paper is to show that these two principles fail completely to help us understand behavior is a very simple model of a duopoly.
In last lecture (AM03), we introduced a simple duopoly model. Two ice-cream vendors buy ice-cream wholesale and can sell at any chosen price in the park. If they have matching prices, they split customers. Under Perfect Competition assumptions, with Full Information and Zero Transaction Costs, if they have different prices, then all customers go to the lower price vendor. Straightforward analysis of this duopoly model leads to the following conclusions:
1. There is a huge amount of genuine uncertainty – probability calculations required for expected utility cannot be made. We cannot know how many people will come to the park on any given day. We cannot forecast the weather conditions, which influence the demand for ice-cream, with any degree of reliability. This means that vendors will adopt rules-of-thumb to make decisions, rather than maximize anything. This leads to the use of evolutionary Agent Based Models as the preferred modeling technique.
2. The strategic calculations we make in the lecture, required by neoclassical theory, are based on EXTREMELY over-simplifying assumptions. In particular, full information eliminates the uncertainty, which in real life, would make estimation of the demand function extremely difficult. Not knowing the demand function, and not knowing the strategy (for price and quantity) that the other vendor will choose, the first vendor cannot possibly calculate profits as a function of his actions. This means that there is no function to “maximize”. The lecture gets over this hurdle by making extremely unrealistic assumptions, to allow us to calculate the demand, so that we can operate in a neoclassical framework. Both sellers know exactly what the other one is doing and exactly how many customers each will get. Furthermore, we show that these are not good approximations, in that when we relax these assumptions, entirely different results emerge.
3. The relation between my actions, and their consequences, is mediated by an uncertain environment (weather, number of people), and by a strategic reaction (what the other guy will do). Economists use a revealed preference argument (due to Savage, Ramsey, De-Finetti) that “uncertainty” (horse races) can be reduced to “risk” (dice rolls). This allows them to use subjective Expected utility theory to create target function to maximize in conditions of uncertainty. However, I have shown elsewhere that this reduction is not legitimate. The standard Dutch book argument use to reduce uncertainty to risk is wrong. This material is not covered in the lecture; see my paper on “Subjective Probability Does Not Exist”.
4. By changing the parameters (the fixed costs, variable costs, demand) we can get NO equilibria, unstable equilibria, and multiple equilibria. Knowledge of equilibria does not tell us anything about what will happen in the real world. What is all important for understanding behavior of dynamic systems is the disequilibrium dynamics: how do the vendors behave when out of equilibrium. It is this behavior that determines what will happen – convergence to equilibrium, divergence away from equilibrium, or continuous cycling between multiple equilibrium.
5. In particular, if we relax the assumptions of full information and zero transaction costs, we find multiple equilibria, including some at which the two vendors charge different prices. This violates the law of one price – the two sellers are identical and selling the identical good, but they charge different prices for it. This is based on the reasonable assumption that even if one vendor charges a higher price, not all customers will leave him to go find the cheaper seller. Either customers do not have information, or they incur transaction costs by walking to the next stall. This shows the extreme sensitivity of supposedly central conclusions of economic theory to the virtually impossible assumptions of full information and zero transaction costs.
6. The typical configuration of costs and profits leads to a Prisoner’s Dilemma in the Duopoly – both parties can profit by cooperation, by agreeing to charge the high monopoly price. However, the individual incentive is to under-cut the price, which leads to complete capture of a smaller profit. If both parties under-cut, then they both share a smaller profit. This is a “social dilemma”, where pursuit of selfish individual incentives leads to loss to both players. This is exactly the opposite of the “Invisible Hand” where pursuit of selfish motives (supposedly) leads to social benefits. Conventional Textbooks mention social dilemmas, but do not point out the conflict with the glorious Invisible Hand, since that would go against the ideological theme of free and unregulated markets creating efficiency.
To summarize, even in very simple real world situations, “maximization” is not possible because there is genuine uncertainty, which cannot be reduced to risk (quantifiable uncertainty). We simply do not know, and cannot calculate, the consequences of our actions, because there are too many other variables which determine this outcome. In addition, as studies of dynamic systems reveal, behavior in such systems is governed by disequilibrium dynamics, and not by the equilibria. In complex systems, study of the equilibria will not reveal any interesting aspects of the behavior, showing the economists must study what happens out of equilibrium to understand how the economy will behave. Thus “maximization” and “equilibrium” are not useful tools to study the behavior of even very simple economic systems. Furthermore, the central teaching that if every is free to maximize, this leads to socially optimal outcomes is directly violated in Prisoner’s dilemma situations where pursuit of individual profits cause harm to society, and even to the individual selfishly pursuing his own profits. Thus the main rhetorical strategy of conventional textbooks is to HIGHLIGHT the polar extreme cases where theory of perfect competition holds, which supports their ideological stance. A vast range of cases which deviate, even slightly, from this PERFECTION, are completely neglected and ignored, because they lead to situations where free markets create bad outcomes. Overcoming market failure requires either government intervention, or utilization of social dimensions of human behavior – humans know how to cooperate, and to sacrifice individual/personal gains for welfare of society. Both of these ideas go against the core ideology of conventional textbooks and hence are not pointed out. Video Lecture of aobut 90minutes is linked below:

A fairly detailed (3500 words) SUMMARY & OUTLINE of lecture is given below:

1. Case of Duopoly (Prisoner’s Dilemma)
We continue our study of the duopoly game from the previous lecture.
There are two possible moves in case of duopoly. If both go to sell 70 units of ice cream then both will get 13,750 and it’s the Mutually Cooperative Pay-off. Now if the new entrant, Y, decides to betray (plays 60 while X is playing 70) then his profit will be 24,000 while X will get nothing. It is called Sucker Pay-off means that X trusted Y but got played as sucker (fool) and get zero.
X       Y 60                            70
60     (X:11,500,Y:11500)       (X:24,000,Y:0)
70     (X:0,Y:24,000)               (X:13750,Y:13750)

How to understand this game according to classical theory?
If X plays 60 then Y should also play 60 as in case of 70 Y will get nothing. Now if X plays 70 even then Y’s response is still 60 as profit will be 24,000 for 60. Thus it is clear that low price 60 is the Dominant Strategy for Y – it is best move for Y, regardless of what X plays. The same holds for X as game is symmetric and best response against all moves of Y is 60. According to standard game theory, both guys will play 60 and both will get profit of 11,500. This situation is called a Social Dilemma because if both people cooperate and plays 70 then both can get higher profit (13,750 each). Economics textbooks teach us about the Invisible Hand: that if everybody acts selfishly then it will be best for society. Here this is proven wrong. If both people act selfishly, they are both hurt, and society is harmed. If both people act cooperatively, in social interest, than BOTH gain. Selfish action not only harms society, it harms the person who takes selfish action.
2. Price War: At Next Move, A New PD Game Emerges
As both players, X and Y, selected 60 move so all higher moves will now have no value and lower strategy now comes in and both players try to undercut their opponent. If this process continues then in second stage the (50, 50) will be the Nash equilibrium as shown in table below. In this way both players will end up at the bottom level of profit due to undercutting each other. This is sometimes called “The Race to the Bottom”
X           Y 60                      50
60       (X:11,500,Y:11500)     (X:0,Y:19500)
50       (X:19500,Y:0)               (X:9250,Y:9250)

3. Conflict between Economic Theory and Reality
There is mismatch between economic theories and what human beings really do. Mengel (2014) did a meta-study (study of studies) in which he found that level of cooperation varies from 3% to 83%. This means that in different type of games there are different levels of cooperation. What explains this difference? She finds that two factors are of importance: Risk and Temptation:
“A Meta-Study on Social Dilemma Games” Friederike Mengel November 6, 2014
4. Risk and Temptation: Drivers of Cooperation
Risk is the price that one pays for trusting other player and if it is high then people don’t cooperate. On the other hand temptation means how much one gain by betraying other and according to Mengel it doesn’t matter as most of the time people are not thinking about betraying others. There are two possible reasons of betraying of which one is to take advantage of the other by deceiving and other is player really want to cooperate but don’t trust the other person. As risk is the main factor it means that mostly people don’t want to betray others according to meta-studies.
There were three games played in our classroom experiment.

  • (B,B)=(5,5)
  • (B,C)=(20,0)
  • (C,B)=(0,20)
  • (C,C)=(15,15)

This case is of low temptation and high risk. Temptation is what you get from betraying when both players are cooperating: Cooperative payoff is 15, You get an additional 5 points if you betray. Risk is what you LOSE from trusting the other player. If you trust other not to betray and play COOPERATE, you end up with 0, so your loss is 15 units of the expected cooperative payoff. In this scenario empirical evidence from Mengel’s meta-study shows that temptation doesn’t matter much but the risk that matters a lot. Since the above matrix of payoffs has high risk, we would expect to see a lot of betrayals, as we found in the classroom experiment in this scenario.
In second case, diagrammed below, there is lower risk because if a player cooperates but is betrayed then he/she loses only 5 points. So more cooperation is expected in this case as per pay-offs in following chart.

  • (B,B)=(5,5)
  • (B,C)=(20,10)
  • (C,B)=(10,20)
  • (C,C)=(15,15)

In the third scenario there is Alternating Betrayals game in which people cooperate to betray each other. As temptation pay-off is very high (50) so smart players will come to an agreement that one time I’ll turn you in and collect the reward and second time you turn me in to collect the reward. In this way both player will have very high reward if they cooperate on multiple rounds.

  • (B,B)=(0,0)
  • (B,C)=(50,0)
  • (C,B)=(0,50)
  • (C,C)=(10,10)

5. Game Theory (Pure)
Initially when game theorists started playing games and their results were completely against classical theory then economists start making excuses that due to playing games in labs and as small amount of money is involved so results cannot be generalized. People don’t play dominant strategy all the time but do cooperate with each other, which economic theory cannot explain. Economic theorists are very reluctant to give up the maximization idea. If everyone always maximizes, then the observed cooperation becomes a “PUZZLE”. Why do people not maximize? Economic theorists argued that the reason may be that game must be played repeatedly. In repeated play, people might learn to cooperate. Accordingly, we study what happens in repeated games.
6. Backward Induction
In a one-shot Prisoner’s Dilemma game, Betrayal is the dominant strategy for both players. Now consider a repeated game, where the same game will be played multiple times. The standard way to solve repeated games is by “backwards induction”. That is, we solve the last round, and then work backwards to the first round in stages. It is easy to calculate the best strategy for the last round of a repeated PD game. We cooperate in the early round because we hope that this will lead to future cooperation by the other party; in every single-shot game, the dominate strategy is always to betray. Once we arrive at the last round in a repeated game, there is no further incentive to cooperate. In the last round, the dominant strategy for both players is to betray. Now both people can calculate optimal strategy for each other. They both note that both are going to play dominant strategy of betrayal in last round. Therefore, both should defect in second-last round. Because cooperation CANNOT create cooperation in the next round. This reasoning goes back to the first round and is called Backward Induction method to solve the game.
Surprisingly, rational optimal calculation gives this as the unique dominant strategy in a repeated PD. Every other solution is dominated. Intuitively, this seems puzzling, because it seems that both people should be able to do better by cooperating, and so rationality should lead to cooperation. This is one place where selfishness of both players hurts both players, and helps us to see that “rationality” as defined by economists, the narrow pursuit of personal gain, is not really rational.
7. Empirical Evidence: People don’t use Backward Induction
Empirical evidence about how human beings behave in games show that people do not use the backward induction strategy to solve games. They think in straightforward way about problem facing them, and postpone thought about what will happen later to a later time (instead of working backwards from the final decision they have to make). Economic theory makes THREE FALSE assumptions about human behavior. The first is that everyone is self-centered and maximize own interest, so they will play dominant strategy of betrayal, instead of the cooperative strategy which works better for both players. The second is that people know how do complex optimization required for multi-stage games. The third, higher level assumption, is that everyone assumes that ALL OTHER people are also self-centered and do maximization at all times. Economic theorists disregard empirical evidence about human behavior. The goal of economic theory is to come up with a model in which everyone is maximizing and yet results match what we see. Thus they keep building fancy models based on beautiful but useless mathematics. The idea that a finite number of repetitions of PD would lead to cooperation as maximizing behavior FAILED to produce a match between theory and observed behavior. So now, current research looks at INFINITE horizon games, in the hope that maximization behavior will lead to cooperation in this (ridiculous) model. In fact, it turns out that cooperation does become possible – that is, with infinite number of games there are pairs of Nash Equilibrium strategies in which both players cooperate. Unfortunately, there are huge numbers of Nash Equilibriums in infinite horizon games – severe MULTIPLE equilibria problem. Because of this, we cannot understand which of the many equilibria will be chosen in reality (although how we can play an infinite horizon game in reality is also a puzzle). So, the infinite horizon case does not really give us any insight into how repeated PD should be played, or is played, by humans.
8. Not Equilibrium but Disequilibrium Dynamics
The major and important lesson we learn from this simple model is that very often there is no equilibrium in real world games, or that the equilibrium is unstable. In these situations, we either start at disequilibrium, or we move to disequilibrium. In order to understand complex systems, it is ESSENTIAL to analyze what happens in disequilibrium. Once disequilibrium take place people follow various rules to try to get out of disequilibrium. No maximization behavior is possible in disequilibrium, because no one knows (including economists) how people will respond to a disequilibrium. Thus, people follow some behavioral RULES, not maximization, to help them make decisions. Depending on which rule they follow, different type of system behaviors can be seen. We may have convergence to any of the multiple equilibria, or divergence, or cycling behavior, where people keep changing moves, without any stable equilibrium ever emerging.
9. MULTIPLE Nash Equilibrium: (30,30) and (40,40)
We show that there are two stable Nash Equilibria in the Ice Cream Duopoly model under consideration, where the wholesale cost of icecream is 25 Rs. For both sellers. The first table below lists the number of customers that P1 and P2 have respectively. At (30, 30), the total number of customers are 1,000 and this is split in half to both players, so both get 500 customers. If P1 increase its price to 40 then he’ll get no customers and same is true for all higher prices. Now if P2 increase its prices 40 as compare to P1 (30) then all customers will go to P1. If price of both is 40 then only 900 customers will be split among them as price increases will drive 100 customers out of the market. Similarly, we can calculate the number of customers that each seller gets at higher prices as shown in table. The key point here is that we are assuming full information and zero transaction costs. We will study what happens when we relax these assumptions in a later lecture. This assumption means that if both vendors offer the same price, they split the overall aggregate demand in half. If either one offers a lower price, he/she gets the full demand, while the other gets ZERO customers. This is of course highly unrealistic.
Now this game can be solved by the Sequential Method. We let one player P1 make an initial move M1. The second player responds by using his optimal strategy M2, which is the best response to M1. Then the first player chooses the best response M3 to M2. Continuing this sequence can converge to a Nash Equilibrium; when there are multiple equilibrium, the one you get to will depend upon the starting point. This shows the importance of history in determining equilibrium. Alternatively, depending on how people respond to each other’s moves, it is possible to have cycling around the equilibrium, with no convergence.
The table below has the possible prices chosen by P2 in the columns. For each responding price by P1, the profits of P1 are listed in the column. This table allows us to pick out the best response by P1 to any strategy (price) chosen by P2. Because the two players are identical and the game is symmetric, we can also use this same table to find the best response by P2 to any move by P1. So we sequence decisions by saying that first P2 chooses a price. Then P1 chooses his best response. This is given in boldface in the second table. This is the highest entry, the one which gives the largest profits to P1. For example, If P2 chooses 30 then P1 will choose 30 as profit is highest (2,000) at 30. At all other prices, P1 will make a loss, since he gets ZERO customers as he has a higher price. So if the game starts at (30,30), then it will stay at (30,30) since both of these strategies are best responses to each other. BUT this is not the whole story, since there is ANOTHER Nash Equilibrium which is better for both players.
If P2 chooses 40 then P1 will choose 40 to get profit of 6250. So (40,40) are both best reponses to each other. Thus if the game starts at (40,40) then this is a stable Nash Equilibrium, and the game will continue to remain in this state. If P2 starts at a higher price – 50,60,70 etc., then P1 will undercut to get all customers. In response, P1 will undercut to get all customers. In this sequential play, each player will undercut until both players are charging 40 – at this price neither player has incentive to undercut the price. According to economic theory, if the players are at the low profit equilibrium, they will stay stuck there forever. In reality, the players can find ways to cooperate to reach the (40,40) equilibrium. With higher levels of cooperation and trust, it is possible for them to reach even higher prices, going back to the monopoly pricing levels at which both players can earn much higher profits. Economic theory based on competition and greed cannot understand or explain such outcomes. However, natural human tendencies towards cooperation and generosity can easily explain these outcomes, if they are made part of a better theory of human behavior.


10. Second Problem: Full Information & Zero Costs
It is ridiculous to say that if one producer changes a price little bit then all the customers will go to low price producer as consumers have full information without any cost.
If we change the assumption little bit as if seller changes the price to 40 then following things happen:
Now there are 900 people willing to purchase instead of 1,000 and remaining 100 customers will go to seller offering at 30. It means now 900 people are willing to purchase at both 30 and 40 so we can split them evenly. In this case 450 (900/2) will go to 40 price seller and 550 (450+100) will go to seller offering at 30.
11. Stiglitz Argument
According to Stiglitz, in perfect competition if one producer increases the price little bit then he will not lose any customer as increase in price is less than transaction cost. But when one producer raise the price then other producers will also raise the price and in this sense we will get second round of equilibrium. And if this process is repeated many times then it may converge to monopolistic equilibrium. The main point is that perfect competition assumption is very fragile as even a very small change in assumption, like a little transaction cast, violate all results of classical theory.
12. Cyclical Equilibrium- Multiple Prices Exist
If there is no law of one price in the market then at (30, 30) the outcome will be same as that of initial case. Now if P1 increases price to 40 then he will get 450 customers (half of 900) and other half and remaining 100 customers will go to P2. Similarly if P1 increases the price to 50 then 400 customers (half of 800) will come to him so P1 losses 50 customers each time he increases the price and opposite is the case for price decrease. Profits are given in second part of the table and are calculated exactly the same way as previous sections. Now to solve the game we need to calculate best responses, shown by bold numbers. If P2 is playing 30 then best response of P1 is 70 to get profit of 13,250. Similarly if P2 increases price to 50 the best response of P1 will still be 70 but after that at 60 P1 switches to 50. Now there are two asymmetric Nash Equilibriums those are (P1:70, P2:50) and (P1:50, P2:70). It means that two sellers will market same good at different prices at same time and the law of one price will no longer works here. Now economic theory cannot predict that which one these two equilibria will emerge; it basically depends on the starting point of the players.
To understand any complex dynamic system we have to consider what happens in disequilibrium, something which goes against core methodological commitments of economists. Yet, even in systems with only one equilibrium, whether or not system arrives at equilibrium depends on the disequilibrium dynamics. In situations where no equilibrium exists, or there are multiple equilibrium, behavior in disequilibrium is crucial to understanding system behavior.


13. No NE or Multiple NE: Study Disequilibrium Dynamics
In perfect competition price is given and firm take decision of how much quantity to sell. Aggregate of this quantity sold by each firm gives the total supply which ultimately influence the price level. The idea of maximization by economists is just impossible to do in real world as it can only be done if everything is known like how many people are going to show up to buy, what the exact price will be and know the exact amount of quantity that the firm is going to sell. But in reality, it is not possible as everybody doesn’t know everything so perfectly. This is just an illustration of Arrow’s puzzle (see Section 16 below): when there is no Walrasian auctioneer, and multiple vendors have choses their own supplies AND prices (multiple prices can co-exist when there is no full information and transaction costs are greater than zero), then no one knows the total supply, and no one will force prices to converge to equilibrium. The nature of the duopoly equilibrium is radically different from the one pictures in textbooks, and also corresponds to what we see in the real world, where identical goods sell at different prices in neighboring stores.
14. Perfect Competition is Fragile- Depends on Unrealistic Assumption
There two methodologies in economics that have recently become popular but not in mainstream economists. One is called the Evolutionary economics in which simulated agents are provided with rules of behavior (instead of maximization) and then we check which rules survive in the long run. Other way is the Behavioral economics in which experiment is run on people to check their behavior in specific situation. In this situation people may learn to cooperate in multiple round games to get the higher pay-offs.
Both the government regulations and cooperation help to get to higher pay-off contrary to what economic theory tells.
15. Theory of Firms
According to the economic theory the main problem of the firm in perfect competition is to set the level of quantity as firm is price taker but it is not true in real world. The problem of finding the quantity at which marginal cost is minimum doesn’t have any solution. Blinder (2000) shows that real firms do 90% of the production under constant or decreasing marginal cost and in this condition the idea of minimizing the marginal cost doesn’t makes sense. So, with increasing production either the marginal cost will remain fix or it will go down unlike economic text books. Issues that arises in real world are completely different from the issues that economists are studying.
16. Arrow’s Puzzle: Who Sets the Prices in Competition?
Arrow say in his article that there is missing piece in every story of economists and one such missing piece is who sets the price in perfect competition. If no one is going to announce the price then from where price is coming? The answer to this illusion is that everyone sets some part of the price. Economists are brainwashed into thinking that law of one price holds but it is not possible either as there hold multiple prices in reality and this is automatically a disequilibrium situation. But economists have to answer to question that what will happen in disequilibrium but Evolutionary and Behavioral theories gives answer to this question.

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