Game Theory

Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue well-defined exogenous objectives (they are rational) and take into account their knowledge or expectations of other decision-makers’ behaviour (they reason strategically). (Osborne and Rubinstein 1994) In short, game theory deals with any problem in which each player’s strategy depends on what the other players do.

Situations involving interdependent decisions arise frequently, in all walks of life. A few examples in which game theory could come in handy include:
● Friends choosing where to go have dinner

● Parents trying to get children to behave

● Commuters deciding how to go to work

● Businesses competing in a market

● Diplomats negotiating a treaty

● Gamblers betting in a card game

All of these situations call for strategic thinking - making use of available information to devise the best plan to achieve one’s objectives. Game theory simply extends this concept to interdependent decisions, in which the options being evaluated are functions of the players’ choices. The appropriate techniques for analyzing interdependent decisions differ significantly from those for individual decisions. Even for strictly competitive games, the goal is simply to identify one’s optimal strategy.

Sequential Games

To analyze a sequential game, first construct a game tree mapping out all of the possibilities. Then follow the basic strategic rule: “look ahead and reason back”:

1. Look ahead to the very last decision, and assume that if it comes to that point, the deciding player will choose his/her optimal outcome (the highest payoff, or otherwise most desirable result).

2. Back up to the second-to-last decision, and assume the next player would choose his/her best outcome, treating the following decision as fixed (because we have already decided what that player will pick if it should come to that).

3. Continue reasoning back in this way until all decisions have been fixed.

The only time you even have to think is if another player makes a “mistake”. Then you must look ahead and reason back again, to see if your optimal strategy has changed.

Simultaneous Games

Turning to simultaneous games, it is immediately apparent that they must be handled differently, because there is not necessarily any last move. Consider a simple, but very famous example, called the Prisoners’ Dilemma: two suspected felons are caught by the police, and interrogated in separate rooms. They are each told the following:

● If you both confess, you will each go to jail for 5 years.

● If only one of you confesses, he gets only 1 year and the other gets 20 years.

● If neither of you confesses, you each get 1 years in jail.

We cannot look ahead and reason back, since neither decision is made first. We just have to consider all possible combinations.                                                                                                    

The game table (also called a payoff matrix) clearly indicates if that the other prisoner confesses, the first prisoner will either get 10 years if he confesses or 25 if he doesn’t. So if the other prisoner confesses, the first would also prefer to confess. If the other prisoner holds out, the first prisoner will get 1 year if he confesses or 3 if he doesn’t, so again he would prefer to confess. And the other prisoner’s reasoning would be identical. There are several notable features in this game. First of all, both players have dominant strategies. A dominant strategy has payoffs such that, regardless of the choices of other players, no other strategy would result in a higher payoff. This greatly simplifies decisions: if you have a dominant strategy, use it, because there is no way to do better. Thus, as we had already determined, both prisoners should confess. Second, both players also have dominated strategies, with payoffs no better than those of at least one other strategy, regardless of the choices of other players. This also simplifies decisions: dominated strategies should never be used, since there is at least one other strategy that will never be worse, and could be better (depending on the choices of other players). A final observation here is that if both prisoners use their optimal strategies (confess), they do not reach an optimal outcome. This is an important theme: maximizing individual welfare does not necessarily aggregate to optimal welfare for a group. Consequently, we see the value of communication. If the two prisoners could only communicate, they could cooperate and agree to hold out so they would both get lighter sentences. But without the possibility of communication, neither can risk it, so both end up worse off.

Game theory is exciting because although the principles are simple, the applications are far-reaching. Interdependent decisions are everywhere, potentially including almost any endeavour in which self-interested agents cooperate and/or compete. Probably the most interesting games involve communication, because so many layers of strategy are possible.

Preferences and Decisions.......

A common assumption in is that ‘‘each individual has stable and coherent preferences’’. In addition, it is often assumed that ‘‘people know their preferences’’. They have the ability or skill in computation to identify calculate the option that maximizes received value, and that they will choose accordingly (Freeman 1993). According to rational choice theory, people have preferences that are related to their behaviour. However, large body of research indicates  that preferences are generally constructed, rather than revealed at the time valuation question is asked. Two major beliefs of the constructive perspective on preferences are that 1. expressions of preference are generally constructed at the time the valuation question is asked and 2 the construction process will be shaped by the interaction between the properties of the human information processing system and the properties of the decision task Payne, Bettman, and Johnson, 1992; Slovic, 1995 ,leading to highly contingent decision behaviour. This view  assumes that people do not have existing well-defined values for many objects; in addition, when asked a valuation question, they will selectively use information that is part of the immediate task description, as well as information that is drawn selectively from memory, to construct a response on the spot. The constructive view also asserts that preferences are not necessarily generated by applying some invariant process such as expected utility maximization; instead, a wide variety of heuristics strategies may be used in constructing a preferential response (Simon, 1955). Individuals may construct preferences because they lack the cognitive resources needed to compute and store well-defined preferences for many situations.

Lets consider consumers behaviour according to "preference construction". This view suggests  the idea that consumer preferences are not well defined but rather constructed in the process of making a choice. So consumer choice  can be manipulated by manipulating the content of the choice or by manipulating which content is the focus of attention ( Shafir, 1993). Scary, when I think about it, most of my purchases were implemented by clever retailers. And I was wondering why I am always broke :)........  

Rational choice and ' Framing effect' and Desicions

     

The major theory of Decision-making under risk is the expected utility model. This model is based on the set of axioms, for example transitivity of preferences, which provide criteria for the rationality of choices. The choices of an individual who conforms to the axioms can be described in terms of utilities of the utilities of various outcomes for that individual. The utility of a risky prospect is equal to the expected utility of its outcomes, obtained by weighting the utility of each possible outcome by its probability. When faced with a choice, rational decision maker will prefer the prospect that offers the highest expected utility (Kahneman & Tversky 1981).However, we need to consider the importance of how the highest expexted utility is presented to us.

 Framing is a cognitive heuristic in which people tend to reach conclusions based on the 'framework' within which a situation was presented.

How does framing effect our decisions? 

A “framing effect” is usually said to occur when equivalent descriptions of a decision problem lead to systematically different decisions. The study of ‘framing effect’ on decisions, led by Benedetto De Martino and Raymond Dolan of University College London, found that participant’s decisions varied due to the framing of the gamble offer.  They used the screen offering two choices. One option was a sure thing, such as "Keep £20" or "Lose £30." The other option was an all-or-nothing gamble. The odds of winning--shown to the subjects as a pie chart--were rigged to provide the same average return as the sure option. In interviews after the experiment, participants said they'd quickly realized that the sure and gamble options were equivalent, and most said that they had split their responses 50-50 between the two choices. But they hadn't. When the sure option was framed as a gain (as in "Keep £20"), subjects played it safe, gambling only 43% of the time on average. If it was framed as a loss, however, they gambled 62% of the time. Thus, individuals' decisions may be altered through manipulation with the framing effect, and the consequences of framing effects may be unavoidable.

                          

Decision making under Risk and Uncertainty

 

Nowadays people make many decisions that involve uncertainty and risk, including important choices like taking insurance, getting mortgage, or medical treatments.Risky decisions, from barely conscious ones when driving ( “ Should I overtake this car? ” ) to carefully deliberated ones about capital investments( “ Do I need to adjust my portfolio weights? ” ). Benjamin Franklin famously stated that the only things certain in life are death and taxes. If anything, the amount of uncertainty in our world has increased between the eighteenth and twenty-first centuries. The economist Frank Knight was the first to make a conceptual distinction between decisions under risk and under uncertainty.  Risk refers to situations where the decision-maker knows with certainty the mathematical probabilities of possible outcomes of choice alternatives.  Uncertainty refers to situations where the likelihood of different outcomes cannot be expressed with any mathematical precision (Weber and Johnson, 2008). 

Daniel Bernouli (1954) found that most of the people don’t tend to maximize expected value. He introduced a game known as St. Petersburg paradox. Asking a question how much would you pay in order to play the game? One toss a coin until it lands on heads. If the coin lands on heads on the first throw, the game ends and you will win £1, if it lands on the heads on the second toss the game ends and you win £2. Every coin toss with tail outcome will double the previous outcome. So from the probabilistic point of view the pay out of the game could be infinitive. However he found that people are nor willing to pay big amount of money to play this game. The expected utility theory indicates that when utility is not favourable or seen as potential losses people will not participate. Simply saying people are more sensitive to looses that gains by showing risk averse behaviour. Nerveless, Kahneman's and Tversky's(1979) prospect theory shows how people handle risk and uncertainty. They argued that people's risky decision will depend on how the problem is formulated. For example the subjective value between gains of £10 and 20£ is greater then the subjective differences between gains of  £110 and  £120.  I think  that the formulation plays a very important role in people decision making.  Our decision will depend on whether the problem was presented to us negatively or positively.

You have 5%  chance  of winning £1000 when betting £100.

      OR

You have 95% chance of loosing £100 when betting to win £1000.

Which one is more appealing to you ?  

PS: I was doing a little research about coin tossing outcomes. If anyone is interested please read this. http://www.codingthewheel.com/archives/the-coin-flip-a-fundamentally-unfair-proposition 

Experiment about coin flip. After I read it, I started to think that if someone would offered me to pay money for  playing the game of coin flipping  like "St. Petersburg Paradox", I would not sign for it :)) 

References
David Hardmam (2009) Judgement and decision making: Psychological perspective. Chichester, UK: BPS-Blackwell
Elke U. Weber & eric J. Johson (2008) Neuroeconomics Decision making and the Brain.   Available on: http://www.mendeley.com/research/decisions-under-uncertainty-psychological-economic-neuroeconomic-explanations-risk-preference/
Last entered 09/11/2011

Bailing and Jailing the Fast and Frugal way. Or are the decisions made in Fast and Frugal way always right ?

This time we had to read articles about decision making in fast and frugal way. The article our group had to read and discuss was ‘Bailing and jailing the Fast and Frugal way’ Dhami and Ayton (2001). The main focus of the article was to review how magistrates make decisions about whether to release defendants on bail based on their previous crimes.  

We would think that magistrates are the people who would carefully consider their decisions, because defendants life and society's well-being depends on how well those decisions are made. Unfortunately, magistrates have to work under constraints such as time pressure. So usually their decisions are being criticized by organizations supporting victims, groups representing defendants and professional agencies such as prosecution agencies. The authors of the article tried to analyze more into depth  how and with what confidence those important decisions are made. They identified the different process models such as the due-process which should work towards reducing crime by minimizing the number of innocent people incorrectly convicted. My attention was drawn to the Fast and Frugal models. For example, the matching heuristic is a simple process model that do not search through all available information, but just through small subset of cues and base decision one cue only. One would think that decisions made based on these heuristics should be inaccurate and poorly made. However, those models are accurate and the evidence for Fast and Frugal models are as good as compensatory integration models.

The results of the study found that majority of magistrates showed inconsistency in their bail decision and they are usually influenced by defended and crime related cues. So simply saying most of the magistrates ‘did not bother’ to look at all the information they were provided with. And in the end their decision depended on their own experience and presumptions about particular individual appearance. However, all magistrates were highly confident that they had made the appropriate decisions. The findings of the article made me wonder whether the justice system is 'Justice' after all and made me to understand how highly social norms and presumptions can influence people's judgments.

The heuristics and biases of judgement under uncertainty. Gigerenze's normative critique of Kahneman and Tversky.

This week our group has been given to read and discus article ‘A reply to Kahneman and Tversky (1996) by Gigerenzer, On Narrow norms and Vague Heuristics’. Thru the discussion of the article we had to form our understanding about the cognitive processes that produce both valid and invalid judgements.

Kahneman and Tversky (1974) described three heuristics that are employed in making judgments under uncertainty; (1) representativeness,  which  is  usually  employed  when  people  are  asked  to  judge  the  probability that  an  object  o r   event  A  belongs  to class or  process  B; (2)availability of  in- stances o r  scenarios, which is  often employed  when  people  are  asked  to  assess the  frequency of  a class or the plausibility  of  a  particular  development;  and (3)  adjustment  from  a n  anchor,  which is usually employed in numerical prediction  when  a  relevant  value  is  available ( Kahneman and Tversky, 1974).  They argued that these  heuristics  are  highly  economical and  usually  effective,  but  they  lead  to systematic  and  predictable  errors.  A better  understanding  of  these  heuristics and  of  the  biases  could  improve judgments  and  decisions in  situations  of  uncertainty. However, Gigerenzer, in his reply to Kahneman and Tversky, argued that some of the biases identified are unstable, for example in some cases their degree can be reduced by asking questions in terms of frequencies rather than in terms of probabilities. Secondly, Gigerenzer argued that, because Kahneman and Tversky’s heuristics are formulated by means of vague, l terms like “representativeness”, the appeal to these heuristics as generators of biases has limited explanatory power. Thirdly, he argued that it may be inappropriate to characterize some of the biases identified by Kahneman and Tversky as “errors” or “fallacies” for three reasons. (a) According to frequentists, no norms are appropriate for single-case judgements, because single-case probabilities are meaningless. (b) Even if single-case probabilities make sense, they need not to be governed by statistical norms because norms are “content-blind” and can conflict with conversational norms. (c) In some cases conflicting statistical norms exist (“statistics does not speak with one voice”) (Vranas, 1999).

Gigerenzer did not neglect the fact that Kahnemans and Tversky’s work gave a big impact to the field of decision making. However, he states:“In place of plausible heuristics that explain everything and nothing- not even conditions that trigger one heuristic rather than another- we will need models that make surprising ( and falsifiable) predictions and that reveal the mental processes that explain both valid and invalid judgment” ( Gigerenzer, 1996). 

References

A. Tversky and D.Kahneman (1974). Judgemnt under Uncertainty: Heuristics and Biases. Science, New Series, Vol. 185, No. 4157. (Sep. 27, 1974), pp. 1124-1131.

Peter B.M. Vranas (1999). Gigerenzer's normative critique of Kahneman and Tversky. Department of Philosophy, The University of Michigam, 2215 Angel Hall, Ann Arbor.