As we learned in the first part of the blog "Strategy - an introduction", strategy is a general plan to achieve an overall goal or vision under uncertain conditions.
In economics, attempts are made to better assess these uncertain conditions by using game theory to predict market movements and interactions between rational actors. Game theory goes back to the mathematician John von Neumann, who in 1928 investigated how parlor games (e.g. chess and "bluffing" in poker) can be solved mathematically by relating the decisions of several participants to one another.
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In 1944, John von Neumann then published the book "Games and Economic Behavior" together with Oskar Morgenstern, in which the theory previously applied to parlor games was now tested on economic issues. In 1950, John Nash extended this theory from zero-sum games, in which the gains and losses of all participants balance each other out, to situations in which all participants can win (Nash equilibrium), even if they do not necessarily improve as a result (prisoner's dilemma).
Why is it useful to study game theory? Because it deals with different strategies on the market, or more precisely, with the behavior of different companies in competition on the market. The individual players are seen as "players" whose decisions are conditionally dependent on the respective behavior of the other.
In order to be able to make decisions and develop strategies themselves, the players must always keep an eye on the actions and reactions of their fellow players. In game theory, a distinction is made between the cooperative version, in which the players can enter into agreements (binding contracts), and the non-cooperative version, in which the players' self-interest is paramount and no binding agreements may be made. The latter is action- and strategy-oriented, which is why we will only focus on this variant here. In order to better represent the probabilities, the normal form is used, which is listed in a bimatrix and is used in static games where all players make their decisions simultaneously. In contrast, the extensive form uses a game tree with branches and is used in dynamic games where all players make their decisions one after the other (www.bwl-lexikon.de).
The Nash equilibrium and the prisoner's dilemma
As already mentioned, a Nash equilibrium exists if no player can improve his situation in the game or on the market by deviating from his strategy alone. Prerequisites for the prediction are a) there can be no collusion between the players (non-cooperative) and b) all players act rationally. The prisoner's dilemma can be used as a classic example here.
After a crime, 2 suspects A and B are arrested and interrogated separately (neither knows whether the other will confess or not). The decisions (to confess or not to confess) are made independently of each other, i.e. practically simultaneously. This is where the normal form comes into play and the considerations can be represented in a matrix.
The following possibilities exist:
> if both deny the crime, both can only be punished with 2 years imprisonment each for minor offenses;
> if both confess, each receives 5 years imprisonment;
> if only one confesses, the confessor receives a reduced sentence of 1 year and the other the full sentence of 10 years.
The situation can be illustrated in the following matrix:
© bwl-lexikon.de
Example reading: 5,5 in the top left-hand box means that both A and B get 5 years if they both confess; similarly, 1,10 in the top right-hand box means that A gets 1 year and B gets 10 years if only A confesses, and so on.
It is obvious from the table that "confessing" is always the best alternative from A's point of view, regardless of what B does: if B also confesses, A gets 5 years (instead of 10 if A does not confess); if B does not confess, A gets 1 year (instead of 2 if A does not confess). A can therefore not improve his position by simply deviating from the "confess" strategy. And vice versa: from B's point of view, it is also better to confess, regardless of what A does. B also cannot improve by deviating from the "confess" strategy alone (www.welt-der-bwl.de/Nash-Gleichgewicht).
So the most likely solution is that both will confess, with the result that both will get five years in prison. Is this the best strategy? No. It is a compromise based on selfishness and represents a poor overall strategy, as there was obviously a failure to agree on tactics in the run-up to the arrest. If this had been the case, the situation would have turned out optimally for both of them, without the need for further coordination. If neither confesses, the result is the fourth matrix field and each prisoner receives only 2 years in prison.
Companies should actually act according to such non-cooperative rules when it comes to pricing, but especially in the case of mineral oil companies, there is always the impression that price agreements are being enforced, particularly before large waves of travel around public holidays. It is the task of the Cartel Office to prevent these agreements and let the "free" market determine the price. However, as long as the profits from price fixing are significantly higher than the threat of penalties, this model will not change.
© Jochen Hirsch
It is also not advisable to turn a non-cooperative, rational price reduction into a non-cooperative, irrational (emotional) price war, which in the worst case can cause massive damage to both companies. This is what happened in Cologne in the 1980s, when Saturn and Schlembach got into each other's hair and began to undercut each other's record prices on a daily basis. In the end, every record was sold for a maximum of DM 1.99. When economic thinking has to take a back seat and only the "art of war" with the aim of destroying the opposing company is in the foreground, there can no longer be any talk of rational action from an economic point of view, because the damage to one's own company is categorically ignored.
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The mutual reaction shown here, depending on the behavior of the competitor(s), is a typical extensive form that can best be represented in a game tree:
© bwl-lexikon.de
The game tree takes into account the time component in the game, as the reaction of company B only arises as a result of the action of company A. At the end of each possible decision branch are the expected results. The first value stands for the expected development of company B, the second for company A. You can see from the numbers at the end (nominal utility values) of each branch that the respective results are dependent on the actions of the other player.
There are many types of game in game theory. In addition to the cooperative and non-cooperative game types, a distinction is also made between the symmetrical and asymmetrical game. A symmetrical game is not dependent on the strategy of one player, but can be influenced by the strategy of another player, such as in the prisoner's dilemma (see above) and the stag hunt. In the stag hunt, the stag can only be killed by both players together. However, if one of the players decides to go for the easier hunt for the hare, which can be done without the second player, the second player goes home empty-handed. There are also zero-sum games, such as poker and chess, in which players cannot increase their resources. As in real life, the "money" is not gone, it simply changes hands/players (redistribution between winner and loser). In the non-zero-sum game, several winners can emerge if - as in the "battle of the sexes" - interests overlap, resulting in a non-zero total. Here, two parties want to do different things (e.g. soccer match vs. concert). However, they agree that they definitely want to spend the evening together because the alternative would be the worst of all outcomes and therefore a compromise is found by mutual agreement (in the matrix, their own favored event would be rated 3, the compromise 2 and separate events 0).
The prisoner's dilemma can also be depicted as a non-cooperative non-zero-sum game, while the battle of the sexes is a cooperative variant. Finally, there is the simultaneous vs. sequential game. As the name suggests, the players act simultaneously in the first variant (e.g. rock, paper, scissors) and sequentially in the second (e.g. chess).
There are different strategies for all game variants. The easiest strategy to determine and predict is the dominant strategy, or more precisely, the strictly dominant strategy. It promises the greatest benefit/profit for the players. To illustrate this, let's use an example from studyflix.de and call the new founder Sven: Sven opens a new company and wants to establish himself on the market against the big competitor in the lemonade business. The question Sven asks himself is whether he should place more advertising, although he does not know what the advertising strategy of the competition will look like. However, the effect of his advertising will also influence the advertising behavior of the competitor company.
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If the competing company assumes that Sven will advertise, it will follow suit and also advertise its product. With advertising, it receives a nominal utility value of ten, or just six. If Sven does not advertise, but the competitor does, its value increases to fifteen. This means that advertising is always the better strategy for the competition, regardless of what Sven decides. Advertising is the strictly dominant strategy for the competitor company. The same applies to our founder and his startup. The same applies here too: Placing advertisements is the strictly dominant strategy. Since the same strategy applies to both companies, it is quite easy to predict what the outcome of the game will be.
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Another example of a dominant strategy occurred during the Cold War. During this time, the mathematician and Nobel Prize winner John von Neumann advised a nuclear first strike because his game theory made this the dominant strategy (WDR Zeitzeichen, 2022). What he completely ignored, however, was the fact that nuclear deterrence is not based on strictly rational considerations, as required by game theory, but on the emotional idea that there will be no winners in a nuclear first strike, only losers. Fear plays a major and not negligible role here compared to pure calculation. What is worrying is that the algorithms of an AI follow exactly the same strict reasoning and logic that game theory demands. And AI is increasingly penetrating all areas of our lives. It has already become an important part of the military. According to The Decoder, 2023, AI is already being used in the military in logistics, reconnaissance, cyberspace and warfare, with autonomous drones already being described as the third revolution in warfare. If AI increasingly takes over our thoughts and actions, we have already lost the game.
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Summary
With 16 (!) Nobel Prizes in economics on the subject of game theory, every scientist realizes how important this "new" branch has become for all of our lives. Yet it is only a mathematical method that attempts to understand the actions of players in a market as a game and to make predictions. And the influence of game theory already extends from mathematics through economics and law, psychology, sociology and computer science to military applications. The very brief insight into the topic described here just scratches the surface of this very complex and constantly growing scientific field. For those who have little access to mathematics or economics, but would still like to learn more about the exciting field of game theory, we recommend the following textbook, which may be getting on in years, but provides explanations without too many mathematical formulas: "Spieltheorie für Einsteiger" by Barry J Dixit and Avinash K Nalebuff, 1995, Schäffer-Poeschel.