Exchange networks, markets and trust

4.1. Evolutionary dynamics

We start where the analysis in section 3 ended and assume that EquationEquation (1)(1) $$p>\frac{c+e}{b+e}\equiv p^{\mathrm{*}}$$(1) is satisfied. All players have left their local networks and deal with each other in the anonymous market. The trustworthy types are honest, while the untrustworthy types are never honest. However, we now open up for the possibility that the two types need not necessarily hold on to their original type and behaviour. We do this by embedding the framework developed above into a model of social evolution in which the trustworthy types ‘compete’ with untrustworthy types. The two types compete in the sense that the players copy the behaviour that is perceived as successful in the population, where the population is made up of trustworthy and untrustworthy types. The role of the evolutionary process is to determine the proportion of the two types in the anonymous market.

The structure of the social evolutionary process builds on the following simplifying assumptions: We start out with a given distribution of the two types, and the players are randomly matched in the anonymous market. Each player simultaneously implements the strategy dictated by their type which produces a payoff to each. The payoff structure is as before and illustrated in Figure 1. $b$ is the payoff for mutual honest behaviour, 0 is the payoff for mutual dishonest behaviour, $a$ is the payoff for being dishonest when meeting a honest player, while $-e$ is the cost of being honest when meeting a dishonest player, where $a>b>0>-e.$

The players play the game independently of their memory of the past, or their expectations of future interactions with either the same or other players. After each round of play, the population distribution changes according to the relative success of the trustworthy and untrustworthy types. Players know the distribution of types in the population, but do not at the onset have any information about their partner type. Equilibrium is defined as stationary of the distribution of the two types.

As before, $p$ is the fraction of trustworthy types ( $T$-types) in the anonymous market, while the remaining $(1-p)$ are untrustworthy types ( $U$-types). At any given time, the growth rate of the proportion of trustworthy types in the population changes according to the following monotonic evolutionary rule, which includes the much-studied replicator dynamics as a special case (Weibull, Citation1995)⁶ (2) $$\dot{p}=\frac{dp}{dt}\mathrm{ }\left\{\mathrm{ }\begin{array}{l}>0\mathrm{ }{V}^T>V^U\\ =0 \mathrm{a}\mathrm{s} {\mathrm{ }V}^T=V^U \hfill \\ <0\mathrm{ }{V}^T<V^U\end{array}\right.\begin{array}{c}\\ \\ \end{array}$$(2) where $V^T$ and $V^U$ are expected payoffs of types $T$ and $U,$ respectively. Hence (2) states that the growth rate of the proportion of trustworthy types in the population is positive or negative depending on whether the expected payoff of trustworthy types is higher or lower than the expected payoff of the untrustworthy types. The population distribution $p$ will be unchanging (reaching an equilibrium) if the expected payoffs for the two types are equal $(V^T=V^U).$⁷

4.2. Market transactions with incomplete type information

Let us briefly recapture the situation mentioned above, that is, the case where all players have left their networks and deal with each other in the anonymous market (condition (1) is satisfied). With probability $p,$ a trustworthy type meets another trustworthy type and gets payoff $b$ and with probability $(1-p),$ a trustworthy type meets an untrustworthy type and gets payoff $–e.$ Two trustworthy types together produce a good result ( $b$), but when a trustworthy type encounter an untrustworthy type, he suffers a loss ( $-e$). The expected payoff to a trustworthy type is $V^T=pb+(1-p)(-e).$ An untrustworthy type is in a much better position. He never suffers the loss of $-e$ since he never acts honest. On the anonymous market an untrustworthy type meets a trustworthy type with probability $p,$ and gets the highest possible payoff $a.$ With probability $(1-p),$ an untrustworthy type meets another untrustworthy type and gets payoff zero. The expected payoff for an untrustworthy type is $V^U=pa+\left(1-p\right)0.$ By comparing $V^T$ and $V^U$ we see that untrustworthy types always fare better than trustworthy types in the anonymous market since they receive a higher payoff no matter the type they encounter. This means that $V^T<V^U$ for all $p\in (0,1).$ Since untrustworthy types get a higher expected payoff, trustworthy types start to turn into dishonest untrustworthy types. According to (2), the fraction of trustworthy types in the population ( $p$) starts to decrease, and this process continues until $p=p^{\mathrm{*}},$ where $p^{\mathrm{*}}$ follows from (1). When $p=p^{\mathrm{*}}$ the players redraw from the anonymously market and start to form networks, where they are protected from dishonest behaviour. Everybody earns a secure payoff of $c.$ We can establish the following result:

Proposition 5:

When players are randomly matched in the anonymous market and lack information about their partner’s type, there is no room for mutual trust relationships in the long run.

4.3. Market transactions with costly type information

The driving force behind the above result is that the trustworthy types are unable to protect themselves from the dishonest behaviour of the untrustworthy types. In reality it is, however, rarely the case that a player is left completely in the dark with respect to information about the reliability of a possible trading partner – even in a large anonymous market. By performing credit checks, investigating criminal records, seeking information about the partner’s reputation, and the likes, it is possible to form an idea about the partner’s reliability. Can such an opportunity for inspection or information gathering stabilise a situation of trust-based transactions in the anonymous market?

In order to study this question in more details, we build on the ideas of Frank (Citation1988) and Güth and Kliemt (Citation2000) that the trustworthy types can obtain information about their partner type through active inspection – and that such inspection reveals the partner type with certainty. Since inspection generally requires time and effort, we assume also that this imposes a cost of $K$ on the trustworthy types. We denote $K$ as the cost of investing in information.

By investing in information, trade on the anonymous market gives the trustworthy types a payoff of $b-K,$ otherwise they get $pb+(1-p)(-e).$⁸ This shows that the trustworthy types have more to gain from investing in information the smaller $p$ is, since this increases the chance of being cheated. If $p=1$ (everybody plays ‘honest’) it never pays to invest in information. At the same time, the gain from investing in information is greater when $K$ is lower. The relationship between the size of $p$ and $K,$ and the decisions to invest in information, can be seen more clearly by noting that the trustworthy types will invest in information if $b-K>pb-(1-p)e,$ or if (3) $$K<(1-p)(b+e)$$(3)

From section 3 (EquationEquation (1)(1) $$p>\frac{c+e}{b+e}\equiv p^{\mathrm{*}}$$(1) ) we know that the minimum value of $p$ required for trade on the anonymous market to take place is $p^{\mathrm{*}}=\frac{c+e}{b+e}.$ By inserting this into (3) we find that the maximum value $K$ can take for the trustworthy types to still be willing to invest in information is given as: (4) $${\mathrm{ }K}^{\text{max}}=b-c$$(4)

EquationEquation (4)(4) $${\mathrm{ }K}^{\text{max}}=b-c$$(4) shows that the maximum $K$ which the trustworthy types are willing to pay for information is determined by the difference between the uncertain payoff from trading on the anonymous market ( $b$) and the certain payoff from staying in a network ( $c$).

Figure 2 illustrates how the decision to invest in information depends on the size of $p\mathrm{ }$ and $K.$ It shows that the trustworthy types are more willing to pay for information with the smaller $p.$ $K=K^{\text{max}}$ is the maximum value $K$ can take for the trustworthy types to still be willing to invest in information. When $p$ increases, the chance of being cheated decreases. The trustworthy types are therefore less willing to pay for information.

Figure 2. The relationship between $\mathrm{p}\mathrm{ }$ and $\mathrm{K}.$

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Figure 2 also shows that for a given $K=K^1,$ the expected gain from investing in information exceeds its cost for $p<p^{\mathrm{\S }},$ where (5) $${\mathrm{ }p}^{\mathrm{\S }}=\frac{b+e-K^1}{b+e}\mathrm{ }$$(5)

If $p>p^{\mathrm{\S }},$ the proportion of trustworthy types on the market is so high that it is not worthwhile to invest in information to identify the partner’s type. It is better to be unconditionally honest. When $p<p^{\mathrm{\S }},$ the opposite holds. The trustworthy types invest in information since the expected gain from information exceeds its cost. It is best to be conditionally honest.

4.4. Evolutionary stability

Having specified the evolutionary process and established the relationship between the cost of information, the population’s composition and the decision to invest in information, we are in a position to say something more about the problem of evolutionary stability.

Figure 3 shows the expected payoff of the two types for a given cost of information, where the solid line illustrates the payoff for the trustworthy types and the broken line illustrates the payoff for the untrustworthy types.

Figure 3. Expected payoff for trustworthy and untrustworthy types.

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In the interval $p^{\mathrm{*}}<p<p^{\mathrm{\S }},$ the trustworthy types invest in information and act honest only with other trustworthy types. The untrustworthy types receive a payoff of zero, while the trustworthy types, by acting honest and paying for information, receive a payoff of $b-K.$ Since the trustworthy types receive a higher payoff ( $V^T>V^U$), untrustworthy types start to turn into honest trustworthy types. According to (2), the fraction of trustworthy types in the population ( $p$) increase, and this process continues until $p=p^{\mathrm{\text{§}}}.$

In the interval $p^{\mathrm{\S }}<p<1,$ the proportion of trustworthy types in the population is so high that the trustworthy types do not find it worthwhile to invest in information. Instead they act unconditionally honest. The untrustworthy types take advantage of this indiscriminate honest behaviour and earn the highest possible payoff ( $a$) when they encounter a trustworthy type (which happens with probability $p$). The expected payoff for the untrustworthy types is $V^U=pa.$ The expected payoff for the trustworthy types is $V^T=pb+\left(1-p\right)(-e).$ Since the untrustworthy types receive a higher payoff ( $V^T<V^U$), trustworthy types start to turn into dishonest untrustworthy types. According to (2), the fraction of trustworthy types in the population ( $p$) decrease, and this process continues until $p=p^{\mathrm{\S }}.$

To sum up: If the proportion of trustworthy types is less than $p^{\mathrm{\S }},$ their number increases. If the proportion of trustworthy types is greater than $p^{\mathrm{\S }},$ their number decreases. $p^{\mathrm{\S }}$ is, in other words, a stable interior evolutionary equilibrium. Small deviations away from $p^{\mathrm{\S }}$ result in a convergence back to $p^{\mathrm{\S }}.$ This stable equilibrium constitutes a mixed society with a proportion $p^{\mathrm{\S }}$ of trustworthy types and a proportion ${(1-p}^{\mathrm{\S }})$ of untrustworthy types. The trustworthy types invest in information and behave honest when they encounter another trustworthy type. The untrustworthy types are never honest. This gives the following result:

Proposition 6:

Mutual trust relationships can emerge and survive in the anonymous market if there exists a possibility to obtain information about the ‘type’ of the possible exchange partner, and this information is not too costly to gather.

It should be noted, however, that the evolutionary equilibrium ${\mathrm{p}}^{\S }$ can be close to 1 depending on the cost of information $(\mathrm{K})$. From (5) it follows that ${\mathrm{p}}^{\S }⟶1 \mathrm{as} \mathrm{K}⟶0$. Hence, the less it costs to obtain information about who is trustworthy, the less room there is for the untrustworthy types. However, as long as $\mathrm{K}>0$, untrustworthy behaviour is not completely eliminated. Honesty dominates the anonymous market, but there is still some element of dishonesty. In other words, trustworthy and untrustworthy types ‘live’ together in a stable evolutionary equilibrium.