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Simulations and mathematics suggest that there always be a Facebook

Professor Wojciech Wiślicki from NCBJ (right) applies minority game strategy to
Professor Wojciech Wiślicki from NCBJ (right) applies minority game strategy to win with MSc Karol Wawrzyniak (source: NCBJ, Grzegorz Krzyżewski)
Why Facebook is a leader in social networking? Is it better to have a wise king than to give power of making decisions to a majority in a less educated crowd of citizens? Is it always profitable to choose the most popular energy carrier? Answers to such questions may be found in computer simulations carried out in Świerk Computing Centre in NCBJ. The research may be very significant for development of the field of mathematics focussed on theory of minority games.

Even if minority games may look like mental exercises of mathematicians, they find many practical applications, for example to model social behaviour patterns and reactions of financial markets, to optimize utilization of power distribution networks, and even to analyse and manage road traffic. Fundamental significance of minority games has attracted interest of a group of young scientists from the CIŚ Świerk Computing Centre in NCBJ. “Results obtained in many computer simulations done by us are not just interesting; we have also found some analytical expression to describe them. That does not happen very often in that field”, said Professor Wojciech Wiślicki from NCBJ.

Multi-agent minority games describe behaviour of systems composed of a large number of units called players or agents. Individual players use different strategies, mutually interact and make their decisions based on their own experience. Game terms are set so that players gain award only when they constitute a minority. Which strategies are the best? When to switch to a different strategy? How the system will react to player actions?

The problems seem pretty abstract. However, in reality we play minority games in our everyday life not even knowing we are playing. People buy stock as soon as bull market starts, but price slumps are inevitable when it turns to a bear market and that time everyone wants to be the first to start selling. You may try to play a minority game even if you keep out of stock exchange: think of driving during peak hours when you try to find an alternative route of an approximately equal distance but known to/used by a lesser number of other drivers.

Karol Wawrzyniak, PhD student in CIŚ and principal investigator of the research is explaining the idea of minority games: “If I am in a minority, I gain. I start to lose as soon as too many have joined me and I am not in a minority any more”.

Classic theory of games assumes that each player knows everything about the game, is fully rational in his/her reasoning, and acts on the basis of deduction. Mixed strategies of Nash, a scientist who became a hero of the famous “Beautiful Mind” film, prove successful in such circumstances. Players who implement mixed strategies base their decisions on probability distributions of random variables. Suppose there are 60 seats in a bar and 120 people want to get in. Players conclude that it will be best to toss a coin. Since that strategy is applied by all players and head/tail probability is fifty/fifty, in effect only about 60 people will show up in the bar. Rules of classical games have been described pretty well by mathematicians, but the problem with them is that probably nobody applies them to make a decision to go to a bar!

Contrary to classical games, in minority games players do not know everything about the game and are reasoning inductively on the basis of their experience. This situation resembles reality. We have no idea how many persons want to visit the bar, we are just thinking: “The last time the bar was overcrowded, may be this time it won’t be?” or “I’m not going since the bar was overcrowded the last time and the next to last time I was there”. There is no static equilibrium (comfortable situation for all players) in such games. Majority is always unhappy and tries to alter their decisions.

“In our research we have analysed dependence of situation in a virtual bar on number of events remembered by players, number of players, and  number of strategies available to them. The rules seem simple, but behaviour of many agents governed by the rules exhibits very complex dynamics” said Wawrzyniak.

A more general version of minority game with more possible objects concerned by decisions of players (e.g. several bars or markets) was also investigated in NCBJ. Each player could first select a market, then decide whether to sell or to by stock on that market. Behaviour of sets with numerous players was simulated numerically. In the initial phase players operated on many markets each offering an identical potential for gains. One might guess that a similar number of players will be attracted by each market also after some time. However, players – even without any personal preferences – have in fact gathered on a single market.

“It turned out that players were attracted by that market, on which some larger fluctuation appeared for the first time”, explains Wawrzyniak. “Social networking Websites grow in a similar way. Plenty of such sites operate at the beginning, but with time people gather at one of them. You can express the idea humorously: simulations and mathematics show that there always be a Facebook”.

Professor Wiślicki notes that it is very interesting to apply results of simulation of minority games to energy markets. “To select a less popular energy carrier might be a beneficial strategy for a country. If all countries invest in sustainable energy sources, nuclear power might be particularly profitable since fuel is going to be relatively cheap and available on such market – at least for some time”.

NCBJ researchers have also shown how to use minority games theory to forecast winning moves. They investigated dependency of forecast accuracy on the number of participating players. It turned out that the largest success is achieved by such group, in which players have transferred their individual strategies to a single leader. Decisions of such a leader having at his/her disposal many strategies are better fit to reality and to a weaker degree attenuated by players implementing individual, usually less flexible strategies.  “The odds are that it is better to have a wise king than a less educated crowd of citizens. The conclusion may be politically incorrect, but this time it is pure mathematics”, laughs Wawrzyniak.

Karol Wawrzyniak started his research on minority games as a PhD student in Interdisciplinary Centre for Mathematical and Computational Modelling of the Warsaw University in cooperation with Ruprecht-Karol University in Heidelberg. Currently he is conducting the research in National Centre for Nuclear Research in Świerk. One of the goals is to develop tools that may help to optimize national policies in regard to future development of power industry in the country.

Pictures

Professor Wojciech Wiślicki from NCBJ (right) applies minority game strategy to