Gaming the System: Using Game Theory to Understand the Promise of Collaboration in Public Service Provision

This piece by Dr Mehdi Shiva and Dr Clare FitzGerald, GO Lab Economist and Research Fellow respectively. They look at game theory and how it can be applied to collaboration in public services.

Working together is better than going it alone 

We live in highly interdependent societies where progress is a collaborative pursuit. Modern examples of collaboration abound, especially from the private sector amongst firms who produce complex products and services. Collaborative relationships emerge between businesses as well as with customers, often with a goal of enhancing market position, lowering transaction costs, and increasing learning (Kogut, 1988). Enabling factors for successful collaborations including the level of trust, coordination, information quality and sharing, and joint problem solving (Monczka et al. 1998, Tomkins, 2001).  

Nevertheless, not every product and activity are the result of fruitful collaboration. The decision to collaborate with external partners is often a fraught one for executives and even collaborations with the best of intentions can go sideways. This is especially true in the public sector, where increasingly people perceive a need for collaboration between government and the private and voluntary sectors but the conditions under which such an approach would yield better results for citizens remain unclear. 

We think game theory provides an interesting theoretical frame for the question of why and when public sector actors collaborate. In this blog, we use game theory to articulate the benefits to society of collaboration in public services and briefly explore why collaborative approaches are not always pursued even when they result in the best outcomes for society.  

Rules of the game: Applying game theory to collaboration 

Game theory is the formal investigation of conflict and collaboration where collaboration is always the optimal solution for collective welfare. It is applicable in situations where two or more parties interact. It shows that a variety of possible factors – such as, say, a lack of trust in others – prevent parties from collaborating with one another. Game theory lays bare the inefficiencies of attempts to maximise personal payoff: doing so makes things worse for everyone else.  

Example: Referral numbers of after school programmes for young people 

Here, we lay out an example for the public sector: youth clubs competing for referrals into their after-school programmes. Public services are unhelpfully fragmented. In the UK, this fragmentation originates with the introduction of the ‘purchaser-provider split’ in the 1980s, when government began contracting out for services. Since then, much has been said about the underwhelming nature of public sector reform brought about through contracting or privatization, where service providers compete for funding, chiefly in the form of government contracts.  

So, picture a small English town (see figure 1) with two youth clubs – West Institution and East Institution – and a population of 200 kids who are eligible and demand their after-school programmes. The two clubs have the capacity to support all 200 kids, provided they work together to share the referrals coming in from local schools. The Local Council, however, is looking to renew their contracts with these youth clubs and is keen to understand which organisation ‘performs best.’ With this added layer of scrutiny, both organisations are considering whether they should spend more on school outreach activities to drive up their referral numbers.  

• Both Institutions collaborate: they manage to cover all the children in their area of convenience. This means each would cover 100 kids, we use the notation (100, 100) for this, where figures represent the number of kids covered by each institute. 
• One Institution collaborates, the other doesn’t: One manages to visit more schools and gets 120 referrals. The added outreach comes at a cost however, which means some resource is directed away from programmes. The other Institution is left with 70 referrals, notated as either (120, 70) or (70, 120). There is a loss of 10 to the society (deadweight).
• Neither Institution collaborates: If neither collaborates, they have to invest their resources (money/staff) in visiting schools out of their territory which result in a loss of 40 units for the society (80, 80). 

Table 1 displays the alternative scenarios wherein either organisation elects to ‘compete’ (i.e., one spends on outreach), both ‘compete’ (i.e., both spend on outreach), or they ‘collaborate’ (i.e., they share details about which schools they are operating in to assure complete coverage).  The numbers below represent the number of children each organisation can support under the given scenario. Where we see competition, an organisation loses the ability to serve 20 children due to the redirection of programmatic funds to outreach. Adding the total of children supported in each scenario reveals that ‘collaborate’ is optimal for the English town: all 200 children receive services. 

The green cell is the optimal solution. The yellow cell displays what happens based on dominant strategy, and is what economists call the Nash Equilibrium (Nash, 1950). It can also be thought of as an outcome where no player has an incentive to deviate unilaterally from the strategy that led to the outcome. Once East and West Institutes decide to compete, neither firm can improve their outcome by changing their approach by themselves.  

Failures of strategy: why collaboration doesn’t happen 

So how can we move from yellow to green? Collaborating would be an optimal strategy for the group collectively, but individual incentives prevent this outcome from being achieved. For example, if East Institute (EI) thought that West Institute (WI) would collaborate, EI would have an incentive to betray and compete with WI rather than to collaborate because their individual payoff would increase to 120 referrals from 100. Same for WI.  

This lack of trust is sensible, but it also drives the selection of suboptimal choices as there is an incentive for the other player to choose to compete to improve their position in the expense of an overall deadweight for the society. One of the main factors that creates this kind of distrust is lack of information, as knowing more about other potential collaborators could help predict a better response for them in the future given the decisions they have made in similar circumstances in the past.  

You will be able to read more about the barriers to collaboration and early thoughts on how to overcome them in the forthcoming LSE blog by the same authors (Shiva and FitzGerald, 2020). We will share the link when it goes live. 

References 

Kogut, B. (1988). Joint ventures: Theoretical and empirical perspectives. Strategic management journal, 9(4), 319-332. 

Monczka, R. M., Petersen, K. J., Handfield, R. B., and Ragatz, G. L. (1998). Success factors in strategic supplier alliances: the buying company perspective. Decision sciences, 29(3), 553-577. 

Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the national academy of sciences, 36(1), 48-49.

Shiva, M., FitzGerald, C. (2020). Why can’t we all just get along?! Barriers to collaboration and early thoughts on how to overcome them in public services. LSE Social Policy Blog.

Tomkins, C. (2001). Interdependencies, trust and information in relationships, alliances and networks. Accounting, organizations and society, 26(2), 161-191.