### 3.1 Altruistic framework with power-based cost and concave utility of throughput

In our scenario, the high complexity of human nature and the surrounding social environment plays a less important role since the cooperation game that we study is limited in time, the identity of the players are hidden, the stakes are relatively low, and the decisions of users are mediated through a programmed device.

So we propose to incorporate in a simple utility function the effect of the external manifestation of altruistic behavior that is a *statistical norm* as termed in [33] or simply ‘what others do’ [28]. To perceive this, the availability of reliable information about the group’s statistical behavior is critical. Our use of the mean idle time per active player to determine the level of altruism in the system is realistic in terms of information availability since it can be easily measured by the different users; though, again, low demand could be mistakenly taken for altruistic behavior and congestion due to a high number of competing users could be mistaken as individually selfish behavior (see discussion on the results shown in Table 1).

Consider a slotted ALOHA random access LAN wherein the *N* ≥ 2 participating nodes control their access probability parameter, *q*. A basic assumption is that nodes’ control actions are based on observations in steady state, i.e., ‘fictitious play’ [34], resulting in a quasi-stationary dynamical system [4, 6, 35] based on the mean throughputs, i.e., for player *i*:

\begin{array}{lcr}{\gamma}_{i}\left(\underset{\xaf}{q}\right)& =& {q}_{i}\prod _{j\ne i}(1-{q}_{j}).\end{array}

Another basic assumption in the following is that the source of a successful transmission is evident to all other participating nodes. We further assume that the degree of altruism *α*_{
i
} of each node *i* depends on the activity of the other users as

\begin{array}{lcr}{\alpha}_{i}\left({\underset{\xaf}{q}}_{-i}\right)& =& \prod _{j\ne i}(1-{q}_{j})\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\frac{{\gamma}_{i}\left(\underset{\xaf}{q}\right)}{{q}_{i}}\\ =& {\gamma}_{i}\left(\underset{\xaf}{q}\right)+\prod _{j}(1-{q}_{j}),\end{array}

where the second term is just the mean idle time of the channel; thus, every node can easily estimate its (dynamic) altruism. By using its control action (strategy), *q*_{
i
}, each *i* seeks to maximize its *net* utility:

\begin{array}{lll}{V}_{i}\left(\underset{\xaf}{q}\right)& =& {C}_{i}log\left({\gamma}_{i}\right(\underset{\xaf}{q}\left)\right)+{A}_{i}{\alpha}_{i}\left({\underset{\xaf}{q}}_{-i}\right){\overline{\gamma}}_{-i}\left(\underset{\xaf}{q}\right)-{M}_{i}{q}_{i}\end{array}

(1)

where the dynamic altruism factor *α* modulates the contribution of the mean service of all other players to the net utility of player *i*,

\begin{array}{lcr}{\overline{\gamma}}_{-i}\left(\underset{\xaf}{q}\right)& =& \frac{1}{N-1}\sum _{j\ne i}{\gamma}_{j}\left(\underset{\xaf}{q}\right);\end{array}

(2)

the utility derived by one’s own throughput is modulated by a concave function [4, 7, 35] as modeled here in the form of a logarithm (for tractability); and we have assumed a power-based cost^{e}*Mq*. Note that because we assume that the source of each successfully transmitted packet is evident to all nodes, each node *i* can easily estimate {\overline{\gamma}}_{-i}. Again, though each player *i* optimizes *V*_{
i
} in a non-cooperative fashion, the game is called altruistic to reflect the second term in (1). In summary, in our model of an altruistic player *i*, benefit (utility) is derived from the success of others ({\overline{\gamma}}_{-i}) and channel idleness (*α*_{
i
}), the latter indicating altruism on the part of others^{f}.

Note that in classical ALOHA, choosing very high (re)transmission parameter *q* results in wasted slots due to interference and wasted transmission power, and choosing very low *q* results in underused (empty) slots. Also note that a single-play slotted ALOHA game between two identical players is similar to the game *chicken*. If *ξ* < 1 is the cost of transmission and the (normalized) payoff of successful transmission is 1, then the Table 1 gives net payoffs for collective action (transmit (Tx) or not) by the players (P1,P2).

The single-play game has three Nash equilibria: two ‘pure’ strategies, (Tx, no-Tx) and (no-Tx, Tx), and one mixed strategy: Tx with probability *q*^{∗} (and no Tx with probability 1 − *q*^{∗}), where *q*^{∗} = 1 − *ξ* jointly minimizes the expected net gains, (1 − *ξ*)*q*_{
k
}(1 − *q*_{3 − k}) − *ξ* *q*_{
k
}*q*_{3 −k}, of players *k* ∈ {1, 2}.

In the following, we consider an *iterated* version of this game where players pursue mixed strategies based on observations of throughput *γ*_{
i
} observed in steady state.

Note that *if* we further assume that nodes are aware of the *C*, *M* parameters of other nodes, then we can replace \overline{\gamma} with the net utility of the other players as in [22] (particularly for throughput-based costs *M* *γ*).

**Proposition 3.1.** If the game is synchronous play and all users *i* have the same (normalized) parameters

\begin{array}{lcr}c:={C}_{i}/{M}_{i}<1& \mathit{\text{and}}& a:={A}_{i}/{M}_{i},\end{array}

then there is a symmetric Nash equilibrium {\underset{\xaf}{q}}^{\ast}={q}^{\ast}\underset{\xaf}{1}, where 0 < *q*^{∗} < 1 is a solution to

\begin{array}{l}f\left(q\right)\phantom{\rule{.5em}{0ex}}:=\phantom{\rule{.5em}{0ex}}a{q}^{2}{(1-q)}^{2N-3}+q-c=0.\end{array}

(3)

#### Proof

When *q*_{
i
} = *q* for all *i*, the first-order necessary conditions of a symmetric Nash equilibrium,

\begin{array}{lcr}0& =& \frac{\partial {V}_{i}}{\partial {q}_{i}}\left(q\underset{\xaf}{1}\right)\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}-\frac{M}{q}f\left(q\right),\end{array}

i.e., equivalent to (3). Note that *f*(0) = − *c* < 0 and *f*(1) = 1 − *c* > 0, the latter by hypothesis. So, by the continuity of *f* and the intermediate value theorem, a root of *f* exists in (0,1).

All such solutions {q}^{\ast}\underset{\xaf}{1} correspond to Nash equilibria because {\partial}^{2}{V}_{i}\left(\underset{\xaf}{q}\right)/\partial {q}_{i}^{2}=-{C}_{i}/{q}_{i}^{2}<0 for all i,\underset{\xaf}{q}. □

The following corollary is immediate.

**Corollary 3.1.** There is a unique symmetric Nash equilibrium point (NEP) if min*q* ∈ (0, 1)*f*^{′}(*q*) > 0 (i.e., *f* is strictly increasing), a condition on parameters *N* and *a*.

Note that there may be non-symmetric Nash equilibria in these games, even for the case of homogeneous users, e.g., [36]. Also, it is well known that Nash equilibria of iterative games are not necessarily asymptotically stable, e.g., [37–39]. In [4, 35] for a slotted ALOHA game with throughput-based costs *M* *γ*, using a Lyapunov function for arbitrary *N* ≥ 2 players, a non-cooperative two-player ALOHA was shown to have two different interior^{g} Nash equilibria, only one of which was locally asymptotically stable (see also [40]).

For stability analysis of our altruistic game, consider the discrete-time (*n*), synchronous-play gradient-ascent dynamics,

\begin{array}{lcr}{q}_{i}\left(n\right)& =& \text{arg}\phantom{\rule{.5em}{0ex}}\underset{{q}_{i}}{\text{max}}\phantom{\rule{.5em}{0ex}}{V}_{i}({q}_{i};{\underset{\xaf}{q}}_{-i}(n-1\left)\right)\phantom{\rule{1em}{0ex}}\forall i.\end{array}

(4)

In a distributed system,^{h} the corresponding continuous-time Jacobi iteration approximation is

\begin{array}{lcr}{\stackrel{\u0307}{q}}_{i}\left(t\right)& =& \frac{\partial {V}_{i}}{\partial {q}_{i}}\left(\underset{\xaf}{q}\right(t\left)\right)\phantom{\rule{1em}{0ex}}\forall i\end{array}

(5)

and is motivated when players take small steps toward their currently optimal response, i.e., better-response dynamics [41]. That is to say, for positive step-size, *ε* ≪ 1 (5) approximates the discrete-time better-response dynamics,

\begin{array}{lcr}{q}_{i}\left(n\right)& =& {q}_{i}(n-1)+\epsilon \frac{\partial {V}_{i}}{\partial {q}_{i}}\left(\underset{\xaf}{q}\right(n\left)\right)\phantom{\rule{1em}{0ex}}\forall i,\end{array}

(6)

which is a kind of distributed gradient ascent. The Jacobi iteration is also motivated by the desire to take small steps to avoid regions of attraction of undesirable boundary NEPs, particularly those corresponding to the capture strategy (*q*_{
i
} = 1 for some *i*). Note that when more than one player selects this strategy, the result is a bad outcome for the game *chicken* or a deadlocked ‘tragedy of the commons.’ Additionally, the players avoid the opt-out strategy (*q*_{
i
} = 0 for some *i*). In summary, (6) represents a repeated game in which players adjust their transmission parameters *q*_{
i
} to (locally) maximize their net utility *V*_{
i
}.

To find conditions on the parameters of net utilities (1) for local stability of the equilibria, we can apply the Hartman-Grobman theorem [42] to (5), i.e., to check that the Jacobian is negative definite. The following proposition uses the conditions of [43] for stability (and uniqueness) for a special case.

**Proposition 3.2.** In the case where players have the same parameters *C* and *A*, the symmetric NEP {q}^{\ast}\underset{\xaf}{1} is locally asymptotically stable under the dynamics in (5) when the normalized parameters satisfy

\begin{array}{lcr}C& >& 2(N-1)A.\end{array}

(7)

#### Proof

By [43], the result follows if the symmetric *N* × *N* matrix H\left(\underset{\xaf}{q}\right) is negative definite, where

\begin{array}{lcr}{H}_{\mathit{\text{ij}}}& =& \frac{{\partial}^{2}{V}_{i}}{\partial {q}_{i}\partial {q}_{j}}+\frac{{\partial}^{2}{V}_{j}}{\partial {q}_{j}\partial {q}_{i}}.\end{array}

First note that for all *i*,

\begin{array}{lcr}{H}_{\mathit{\text{ii}}}\left(\underset{\xaf}{q}\right)& =& -\frac{C}{{q}_{i}^{2}}\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}-C.\end{array}

For *l* ≠ *i*,

\begin{array}{ll}\frac{{\partial}^{2}{V}_{i}}{\partial {q}_{i}\partial {q}_{l}}& =\frac{\partial}{\partial {q}_{l}}\left(\frac{C}{{q}_{i}}-\mathrm{A\alpha}\left({\underset{\xaf}{q}}_{-i}\right)\frac{1}{N-1}\sum _{j\ne i}{q}_{j}\prod _{k\ne i,j}(1-{q}_{k})\right)\\ =A\prod _{j\ne i,l}(1-{q}_{j})\frac{1}{N-1}\sum _{j\ne i}{q}_{j}\prod _{k\ne i,j}(1-{q}_{k})\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\mathrm{A\alpha}\left({\underset{\xaf}{q}}_{-i}\right)\frac{1}{N-1}\left(\sum _{j\ne i,l}{q}_{j}\prod _{k\ne i,j,l}(1-{q}_{k})\right.\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\left(\right)close=")">-\prod _{k\ne i,l}(1-{q}_{k})& .\end{array}\n

Now because 0 < *q*_{
i
} < 1 for all *i* and the triangle inequality,

\begin{array}{l}\left|{H}_{\mathit{\text{ij}}}\right(\underset{\xaf}{q}\left)\right|\le 2A\forall j\ne i.\end{array}

So, by the Gershgorin circle (disc) theorem (see p. 344 of [44]), all of H\left(\underset{\xaf}{q}\right)’s eigenvalues are less than −*C* + (*N* − 1)2*A*. So, if (7) holds, then all the eigenvalues of H\left(\underset{\xaf}{q}\right) are negative, and so H\left(\underset{\xaf}{q}\right) is negative definite. □

### 3.2 The marginal effect of altruism

In this section, we will write *q*^{∗} (of the symmetric NEP {q}^{\ast}\underset{\xaf}{1} in symmetric users case) as a function of the normalized altruism parameter *a* := *A*/*M*, *q*^{∗}(*a*). Note that *q*^{∗}(0) = *c* := *C*/*M*.

Recall that the total throughput for slotted ALOHA, *N* *c*(1 − *c*)^{N−1}, is maximal when *c* = 1/*N*. The maximum total throughput is (1 − 1/*N*)^{N−1}≈e^{−1} for large *N*, i.e., the maximum throughput per player is 1 / (*N* e) in this *cooperative* setting *without* networking costs.

So, if *c* > 1/*N*, i.e., total throughput is less than e^{−1} because of excessive demand (overloaded system), then a marginal increase in altruism from 0 (0 < *a* ≪ 1) will cause a marginal decrease in *q*^{∗}*↓* 1 / *N*, resulting in an increase in throughput per user *γ* *↑* 1 / (*N* e). Also, if *c* < 1/*N*, i.e., total throughput is less than e^{−1} because of a lack of demand (an underloaded system), then a marginal increase in altruism from 0 will again cause a marginal decrease in *q*^{∗}, but here resulting in a decrease in throughput *γ* (further away from the optimum e^{−1}). See Section 5.4.