# Category: Game theory

1) The number of first-year students in the Netherlands has soared from 105 000 in 2000 to 135 000 in 2011. The 30% increase is a direct result of government policy which links university funding with student numbers. In some programs in the country, student numbers have more than doubled during the last five years. Everyone is encouraged to enter the university system. 2) In the general case, there is no selection at the gate. Students cannot be refused to enter a program. 3) Now, the government’s objectives are to reduce the number of first-year drop-outs  and slash the number of students who do not graduate within four years. Both objectives are being supported by financial incentives and penalties for the universities. Something’s gotta give. I wonder what… P.S. ‘Solve for the equilibrium’ is the title of a rubric from Marginal Revolution.

The Prisoner’s Dilemma (PD) is the paradigmatic scientific model to understand human cooperation. You would think that after several decennia of analyzing this deceivingly simple game, nothing new can be learned. Not quite. This new paper discovers a whole new class of strategies that provide a unilateral advantage to the players using them in playing the repeated version of the game. In effect, using these strategies one can force the opponent to any score one desires. The familiar tit-for-tat strategy, which so far had been assumed to be the optimal way of playing the repeated game, appears to be just the tip of an iceberg of ‘zero determinant’ strategies which ‘enforce a linear relationship between the two players’ scores’. This is huge and people have already started to discuss the implications. But what puzzles me is the following: The search for an optimal way to play the repeated PD has been going on at least since the 1980s. The best strategies have been sought analytically, and through simulation (see Robert Axelrod’s iterated PD tournaments). And yet nobody discovered or stumbled upon ‘zero determinant’ strategies for more than 30 years of dedicated research. So can we expect a rational but not omnipotent actor to use these strategies? I think the formal answer needs to be ‘yes’ – a rational actor plays the game in the most advantageous way for his/her interests and if zero determinant strategies provide en edge, then he/she needs to (and is expected and predicted to) play these. The alternative would be to impose some limitations to the…

The latest issue of Political Research Quarterly has an interesting and important exchange about the use of game theory to understand the effectiveness of torture for eliciting truthful information. In this post I summarize the discussion, which is quite instructive for illustrating the prejudices and misunderstandings people have about the role and utility of game theory as a tool to gain insights into the social world. In the original article, Schiemann builds a strategic incomplete-information game between a detainee (who can either posses valuable information or not, and be either ‘strong’ or ‘weak’) and a state which can be either ‘pragmatic’ (using torture only for valuable information) or ‘sadistic’ (torturing in all circumstances). There are two additional parameters capturing uncertainty about the value and completeness of the information provided by the detainee, and two styles of interrogation (providing leading evidence or not). The article then proceeds to identify the equilibria of the game, which turn out to be quite a few (six), and quite different – in some, truthful information is provided while in others, not; in some, torture is applied while in others, not; etc…. At this point you will be excused for wondering what’s the point of the formal modeling if it only shows that, depending on the parameters, different things are possible. Schiemann, however, makes a brilliant move by comparing each of these equilibria to some minimal normative standards that proponents of torture claim to uphold – namely, that torture should not be used on detainees who have provided all their information, that transmitted information should be generally reliable, and that in all cases only…

The plans for a referendum on Scottish independence offer a nice opportunity for applying spatial analysis. The latest point of contestation is whether a third option (enhanced devolution) should be offered to the voters in addition to the ‘Yes’ and ‘No’. The UK government is against including the third option, a Scottish movement is strongly in favor, and the major advocate of the independence camp Alex Salmond is undecided (as far as I can tell). Assuming that the government in London prefers Scotland to remain in the UK (and enhanced devolution to full independence), why do they oppose the inclusion of the third option in the referendum? That would only make sense if the UK government believes that more people would vote ‘No’ to independence when faced with the choice between the two extremes. At the same time, proponents of full independence will be better off including the third option only if they believe that they will lose a Yes/No referendum. Trying to check the current estimates of support for independence, however, does not lead to a straightforward answer. According to Wikipedia, the latest poll conducted in September 2011 places the two camps practically dead-even – 39% say they would vote ‘Yes’ and 38% say they would vote ‘No’. According to the betting markets on the other hand, Scottish independence in the near future doesn’t stand quite a chance. Obviously, London trusts the betting markets more than the polls. With the decision to oppose a third option in an eventual referendum, the UK…

Here is a puzzle: You meet a real estate agent for a property you are interested in. The house has an asking prize and you haven’t made any offers yet. The realtor mentions casually that she has just had an offer for the house which she has rejected. Would you ask what the offer was? Would the realtor tell you? Is it a fair question to ask? (obviously, the realtor is under no obligation to reveal the truth value of the rejected offer and there is no way for me to verify the answer).

Here is a formalized description of the problem: the Seller adn the Buyer can be each of two types – High or Low.  High Buyers and Sellers prefer High Deal to No Deal no Low Deal, and Low Buyers and Sellers prefer High Deal to Low Deal to No Deal. First, the Seller announces whether she has rejected a Very low or a Moderate offer. If a Moderate offer has been (announced as) rejected, the Buyer can make either a High offer (which all Sellers accept) or No offer which ends the game. If a Very low offer has been (announced as) rejected, the Buyer can make a Low offer, No offer or a High offer (the latter two end the game). If a Low offer has been made, the Seller can either Accept or Reject it. In the case of rejection the Buyer can make a High offer or No offer – both actions end the game. Here is the game tree.

Essentially, by making an announcement that she has rejected a Moderate offer the Seller credibly commits to reject any Low offers. Importantly, Buyers suffer a cost from a rejected offer (which is realistic given the costs of the compulsory technical surveys one has to do before an offer). There is no penalty for a late deal (no time discounting). The game is of two-sided incomplete information – neither the Buyers nor the Sellers know the type of the opponent. So the questions:

1) Should you ask what the rejected offer was?
2) Should the realtor (the Seller) tell you?
3) Would the answer (announcement) of the Seller be informative?
4) Does the Seller do better under this game or a game with no signal (announcement)?
5) Does the Buyer do better under this game or a game with no signal?
6) Is this game Pareto-improving under any circumstances?

My answers are after the fold.