This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Many of the issues we face as a society are made more problematic by the rapidly changing context in which important decisions are made. For example buying a petrol powered car is most advantageous when there are many petrol pumps providing cheap petrol whereas buying an electric car is most advantageous when there are many electrical recharge points or high capacity batteries available. Such collective decision-making is often studied using economic game theory where the focus is on how individuals might reach an agreement regarding the supply and demand for the different energy types. But even if the two parties find a mutually agreeable strategy, as technology and costs change over time, for example through cheaper and more efficient batteries and a more accurate pricing of the total cost of oil consumption, so too do the incentives for the choices buyers and sellers make, the result of which can be the stranding of an industry or even a whole economy on an island of inefficient outcomes. In this article we consider the issue of how changes in the underlying incentives can move us from an optimal economy to a sub-optimal economy while at the same time making it impossible to collectively navigate our way to a better strategy without forcing us to pass through a socially undesirable “tipping point”. We show that different perturbations to underlying incentives results in the creation or destruction of “strategic islands” isolated by disruptive transitions between strategies. The significant result in this work is the illustration that an economy that remains strategically stationary can over time become stranded in a suboptimal outcome from which there is no easy way to put the economy on a path to better outcomes without going through an economic tipping point.

One of the most important contributions to the theoretical foundations of the social sciences in recent years has been the introduction of methods from physics, and particularly statistical mechanics, in order to model large scale human social behaviour. The field is vast and includes complex network theory [

The application of these techniques to economic game theory allows us to construct formal models of the simplified interactions between people when individual rewards are dependent on joint choices. The appeal of this as an approach is seen in the number of fields to which these ideas have been applied beyond the micro-economics from which it was borne: evolutionary genetics [

One of the first and most important results in game theory is that of the Nash equilibrium [

While it is common to discuss game theory in terms of two or more people or players, we often have in mind a more general situation in which each player is replaced by two sub-populations of economic agents who make their decisions independently and based on the incentives for the choices that characterise each sub-population. To keep a concrete example in mind we imagine a highly abstract sub-population of car buyers who can choose between electric powered cars or petrol powered cars and a sub-population of energy providers who can sell either electricity through recharge outlets or petrol through traditional petrol stations.

Using this framework of non-cooperative game theory the article is divided into the following sections. Section 2 introduces the necessary notation and ideas from classical non-cooperative game theory. Readers familiar with game theory may choose to skim this section noting only the notation that will be used later. Section 3 introduces a stochastic form of game theory equilibrium called the Quantal Response Equilibrium (QRE) and two approaches to its role as an equilibrium solution are reviewed and related to the Maximum Entropy technique of Jaynes [

The notion of stochastic dynamics and imperfect decision-making forms the basis of the work in this article and is introduced in Section 3, but first we introduce the Nash equilibrium [

The goal of game theory is to model the choices adopted by players based on specific variations in the concept of “rational choice”. For example Nash described a type of rationality in which each player can choose a distribution over their options, _{c}_{c}_{r}_{r}

where

such that neither player can increase their expected utility by unilaterally changing their strategy. Two players who maximise their expected utilities in this fashion will arrive at a Nash equilibrium. Pure strategies are those in which _{c}

Instead of using the probabilities as shown above, it will be useful to frame what follows in terms of a _{r}_{c}

The expected utilities can then be expressed as a polynomial in player strategies _{r}_{c}

We will also make use of the _{r}_{c}

We will also use the difference in the conditional expected utilities for two choice games, given by:

In the next section we introduce the Quantal Response Equilibrium, an alternative equilibrium containing the Nash Equilibrium as a limiting case.

While the Nash equilibrium has played a key role in modern game theory, it has a number of shortcomings when used to describe the observed behaviour of decision-makers [

We begin with the entropy of a probability distribution _{i}

The

The MaxEnt technique applied to game theory begins with the notion of finding the distribution
_{x}_{i}_{x}_{x}

subject to the constraints:

Such a constrained optimisation problem can be solved by forming the Lagrangian
_{x}_{{}_{px}_{x}_{x}

where the
_{r,c}_{r}_{c}

This is the hyperbolic behavioural form of

The QRE has been used as a model of bounded rationality [_{r}_{c}_{r}_{c}_{r}_{c}_{x}_{r}_{c}

There are at least three ways in which the QRE can be derived, the original approach of McKelvey and Palfrey and MaxEnt have already been discussed and these are often specifically interpreted as choice models for individual economic agents. The third is based on the fixed point solution to a Fokker–Planck equation (FPE) phrased in terms of stochastic partial differential equations applied to entire populations of incentivised agents [

A few comments are in order to provide an interpretation of the

From this perspective, there is good reason to suggest that

Next we illustrate the QRE and how variations in behavioural parameters (the

The Prisoners’ Dilemma is a quintessential example of the counter-intuitive results that are possible in game theory. The story of the game is that two criminals are picked up for the same crime and placed in two separate cells so that they cannot communicate and both suspects are offered the same deal: If you both choose to remain silent as to who committed the crime both suspects will get 1 year in jail, if one accuses the other while they in turn claim innocence, the accuser goes free and the accused receives a 3 year jail term, but if they both accuse each other of the crime they will each receive a 2 year jail term. The situation is depicted in the payoff matrices (_{x}_{x}

The expected utilities for the two players are:
_{r,c}_{x}_{i}_{x}_{x}_{x}_{r}_{c}_{x}_{x}

The equivalent analysis for the QRE gives a similar result, except that parametric variations result in paths that remain in equilibrium. _{x}_{r}_{r}_{c}_{c}_{x}

So the players still have an incentive to increase their respective _{x}_{x}

One of the most important contributions to the theoretical foundations of the social sciences in recent years has been the introduction of methods from physics, and particularly statistical mechanics, in order to model large scale human behaviour. Amongst the many popular models the canonical 2-D Ising model and its mean field approximation have been rigorously studied as an analogy to social interactions [

where

The QRE (_{r}_{c}_{r}_{c}_{r}_{c}_{r}_{c}

where

_{x}_{x}

These systems have been studied extensively but new results still appear regularly [

This work focuses on the

In order to illustrate some of the more unusual results that are possible with equilibrium solutions to the QRE, the following examples will focus on a perturbed game matrix given by:

and the expected utilities:

where _{r}_{c}_{x}_{c}_{r}_{x}_{r}_{c}

and

The QRE has an important characteristic called the _{row}_{col}_{row}_{col}_{row}_{row}_{col}

By changing matrices (_{r}_{c}_{x}_{r}_{c}

_{r}_{c}

The _{x}

The true threat is not in the actual changes in price though, seen as drops in the expected utilities shown on the bottom row of _{row}_{col}

Taking the previous example of strategic islands as the starting point, we next consider what happens if the utility for the chosen strategy continues to deteriorate. This is illustrated in _{r}_{c}_{r}_{c}

It may be possible to try and “outrun” the encroaching bifurcation by adjusting the _{r}_{c}_{r}_{c}

Again we consider a given equilibrium solution for a _{r}_{c}_{row}_{col}

This result is to be expected given the structure of the utility matrices where the different _{col}_{row}_{col}_{c}_{row}_{r}

We can also use the McKelvey and Palfrey tracing procedure for the principal branch in order to understand some of the properties of this equilibrium surface. Setting _{c}_{r}_{col}_{r}_{r}_{c}_{row}_{r}_{r}_{c}_{col}_{row}_{r}_{c}

Finally, in _{r}_{c}

This article has focused on the qualitative features of the Quantal Response Equilibrium, an economic model used to represent imperfect economic decisions made in a social context. The emphasis has been on the “tipping points” of this model because of the incredible impact such discontinuous transitions can have on economies and society in general [

There are three key findings that have not, at least within the framework of game theory, previously been examined as consequences of the QRE and boundedly rational decisions across an economy. The first is that the changing environment can shift a previously good strategy for two populations of decision makers to a sub-optimal outcome for both populations. This occurs when a previously good strategy has a progressively deteriorating utility. Moreover, this strategy can become isolated from the rest of the strategy space in the sense that other regions, and in particular better strategic regions, are cut-off such that they are not smoothly attainable from the current strategy.

To use the car fuel analogy, a population that buys a majority of petrol powered cars when petrol prices are low and electricity supplied for cars is expensive is an economically sound strategy. But as the cost of petrol increases while the supply of cheap electricity to fuel stations increases, then what was previously a good strategy (petrol cars and petrol stations) may become sub-optimal compared to electric cars and electric fuel stations. In order to achieve this better strategy though it may be the case that the car and energy sectors need to pass through a tipping point where both industry sectors suddenly collapse to a new equilibrium of electric cars and electric power outlets given that the previous equilibrium is not sustainable. The result could be the sudden under-utilisation of resources such as highly trained staff and manufacturing facilities while at the same time leaving industries technologically unprepared to produce new cars and fuel them, despite this being the better strategy. By far a

The second significant finding is closely related to the first: even if the current strategy is maintained, further degradation in the utility of the existing strategy may result in the equilibrium strategy “evaporating”, leading to a different tipping point not produced by trying to navigate the space of equilibrium solutions but instead caused by the decreasing quality of the current strategy through a progressively degraded utility of that strategy.

The third significant finding is that different changes in the underlying incentive structures result in different outcomes in terms of whether or not the existing strategy is optimal or sub-optimal. The analogy in the car industry example is that one industry is preferentially incentivised over another. In this case we can imagine that oil might become more expensive to extract and transport reducing the overall profitability of petrol stations whereas the price of petrol powered cars might be subsidised through a government intervention program, resulting in cheaper petrol cars relative to electric cars. The result can be a sub-optimal outcome for the energy sector but an optimal outcome (perhaps artificially so) for the car industry.

M. Harŕe initiated the idea, wrote the software scripts and wrote the article. T. Bossomaier assisted in framing the results and co-wrote the article. Both of the authors have read and approved the final manuscript.

The authors declare there are no conflict of interest.

Plot of the QRE fixed point solutions of the Prisoner’s Dilemma game (matrices (

Plot of fixed point solutions of

The tracing procedure and tipping points illustrated. The left figure is the QRE surface as given by _{r}_{c}_{row}_{row}_{row}_{row}_{row}_{row}_{row}

Perturbed QRE solutions for _{c}_{r}_{c}_{r}

The game as in table (_{r}_{c}_{r}_{c}_{col}_{row}

Perturbed QRE solutions for both players for _{r}_{c}

Perturbed QRE solutions for (from left to right in the figure): {_{r}_{c}_{r}_{c}_{r}_{c}